domain: math nlin

text:
Olivier Marchal
email: 1olivier.marchal@univ-st-etienne.fr

Mohamad Alameddine
email: 2mohamad.alameddine@univ-st-etienne.fr

Isomonodromic and isospectral deformations of meromorphic connections: the sl 2 (C) case 1

 in the case of sl 2 (C).

Introduction

Isomonodromic deformations of meromorphic connections have been studied for a long time [START_REF] Picard | Mémoire sur la théorie des fonctions algébriques de deux variables[END_REF][START_REF] Garnier | Solution du problème de Riemann pour les systèmes différentiels linéaires du second ordre[END_REF][START_REF] Painlevé | Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme[END_REF][START_REF] Schlesinger | Über eine klasse von differentialsystemen beliebiger ordnung mit festen kritischen punkten[END_REF][START_REF] Jimbo | Monodromy preserving deformation of linear ordinary differential equations with rational coefficients[END_REF][START_REF] Jimbo | Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and τ -function[END_REF] in relation with the Painlevé property. The main feature of isomonodromic deformations is that they contain an underlying Hamiltonian symplectic structure associated to the Darboux coordinates describing the Lax matrices. This property has been extensively described and proved for a very large class of meromorphic connections [START_REF] Adams | Darboux coordinates and Liouville-Arnold integration in loop algebras[END_REF][START_REF] Adams | Darboux coordinates on coadjoint orbits of Lie algebras[END_REF][START_REF] Hurtubise | On the geometry of isomonodromic deformations[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF] including those studied in the present article. In particular, the case of Fuchsian singularities (i.e. simple poles) has been known since the works of R. Fuchs [START_REF] Fuchs | Sur quelques équations différentielles linéaires du second ordre[END_REF] and B. Gambier [START_REF] Gambier | Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes[END_REF] giving rise to the Lax formulation of the Painlevé 6 equation. On the contrary, the case of irregular singularities (i.e. meromorphic connections with poles of arbitrary orders) is much more difficult to handle and many issues remain opened and is still currently generating active research [START_REF] Harnad | Hamiltonian Dynamics, Classical R-matrices and Isomonodromic Deformations[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF]. There are several complications in the case of irregular singularities. First, the underlying geometry of the moduli space is much more complex to describe because one needs to include Stokes data in addition to the location of poles although there have been recent developments in the understanding of the natural geometric parameters [START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF]. If the existence of an underlying Hamiltonian symplectic structure is known in the case of irregular singularities, getting some general formulas to express the Hamiltonian remains a difficult question.

There are several strategies to tackle irregular singularities. A first option is to start with the well-known case of simple poles and make confluences of these simple poles to generate poles of higher orders. This strategy has been used in many papers [START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Mazzocco | The Hamiltonian structure of the second Painlevé hierarchy[END_REF][START_REF] Chernyakov | Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems[END_REF] with very interesting results developed in [START_REF] Gaiur | Isomonodromic deformations: Confluence, reduction & quantisation[END_REF]. A second strategy is simply to write down the compatibility equations of the Lax system (Cf. [START_REF] Adams | Darboux coordinates on coadjoint orbits of Lie algebras[END_REF][START_REF] Adams | Dual moment maps into loop algebras[END_REF][START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF][START_REF] Chernyakov | Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems[END_REF][START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Fuchs | Sur quelques équations différentielles linéaires du second ordre[END_REF][START_REF] Gaiur | Isomonodromic deformations: Confluence, reduction & quantisation[END_REF][START_REF] Gambier | Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes[END_REF][START_REF] Garnier | Solution du problème de Riemann pour les systèmes différentiels linéaires du second ordre[END_REF][START_REF] Harnad | Dual isomonodromic deformations and moment maps to loop algebras[END_REF][START_REF] Harnad | Hamiltonian Dynamics, Classical R-matrices and Isomonodromic Deformations[END_REF][START_REF] Hurtubise | On the geometry of isomonodromic deformations[END_REF]), which corresponds to a zero-curvature equation for connections and try to solve it to get the complete expression for the Lax matrices and the Hamiltonian system. This strategy was partly solved in [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF] for meromorphic connections in sl 2 (C). Indeed, in this article, the authors provided explicit formulas for the Lax pairs and the Hamiltonian in the oper gauge (i.e. the gauge where the Lax matrix is companion-like) for non-twisted meromorphic connections with poles of arbitrary orders. The main limitation of these results is that they express the Lax pair in the oper gauge and obtain the Hamiltonian structure relatively to coordinates adapted to this gauge which is not the natural geometric gauge. Solving this issue by defining appropriate Darboux coordinates and obtaining explicit expressions in the initial geometric gauge was one of the first motivations for this article that we achieved in Theorems 4.1, 4.2 and 4.3.

Let us finally mention that there exists a third strategy to tackle isomonodromic deformations by considering first isospectral deformations (i.e. deformations that preserve the spectrum of the Lax matrix) and then choose some specific Darboux coordinates so that an additional "isospectral condition"

δ t L(λ) = ∂ λ Ã(λ) (1-1)
is realized [START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF]. The main feature of the additional isospectral condition is that when it is realized the Hamiltonian relatively to the Darboux coordinates identify with the spectral invariants of the Lax matrix (i.e. the coefficients of the expansion of the eigenvalues of the Lax matrix at each pole). This isospectral strategy has been used successfully for many cases such as Fuchsian singularities and all Painlevé cases and the general theory has been set up in [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF] by the Montréal school. Let us mention here that the isospectral condition puts strong constraints on the explicit dependence of the Darboux coordinates relatively to the deformation parameters. For Fuchsian singularities, it turns out that the additional isospectral condition is immediate to solve and that the explicit dependence of the Darboux coordinates relatively to the deformation parameters is trivial. Unfortunately, this observation does not hold for non-Fuchsian singularities and if one can find the explicit dependence of the Darboux coordinates relatively to deformation parameters on a case by case basis for meromorphic connections with singular poles of low orders, the authors of [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF] left the issue of finding Darboux coordinates solving the isospectral condition in a general setting opened. The second motivation of the present article was thus to give an explicit description of the Darboux coordinates solving the isospectral condition in the case of meromorphic connections in sl 2 (C) leading to Theorem 5.5. As expected, the explicit dependence of the Darboux coordinates relatively to deformation parameters is more complicated for non-Fuchsian singularities.

In the end, our final conclusion is that solving directly the isomonodromic compatibility equations or solving the isospectral condition are equivalent problems that share the same level of difficulty for sl 2 (C). Both strategies have interests and technical difficulties but we hope that the present paper shall draw some explicit bridges and dictionary between the two approaches.

2 Meromorphic connections, gauges and Darboux coordinates

Meromorphic connections and irregular times

One of the main results of [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF], which is also obvious from the underlying Poisson structure, is to show that the Hamiltonian structure is the same for meromorphic connections in gl 2 (C) and in sl 2 (C). Thus, we shall restrict, without loss of generality, to meromorphic connections in sl 2 (C) in the rest of the article.

The space of sl 2 (C) meromorphic connections has been studied from many different perspectives. In the present article, we shall mainly follow the point of view of the Montréal group [START_REF] Adams | Darboux coordinates and Liouville-Arnold integration in loop algebras[END_REF][START_REF] Adams | Darboux coordinates on coadjoint orbits of Lie algebras[END_REF] together with some insights from the work of P. Boalch [START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF]. Let us first define the space we shall study.

Definition 2.1 (Space of rational connections). Let n ∈ N be a non-negative integer and {X i } n i=1 be n distinct points in the complex plane. Let us denote R := {∞, X 1 , . . . , X n } and R 0 := {X 1 , . . . , X n }. For any r := (r ∞ , r 1 , . . . , r n ) ∈ (N \ {0}) n+1 , let us define

F R,r := L(λ) = r∞-1 k=1 L[∞,k] λ k-1 + n s=1 rs-1 k=0 L[Xs,k] (λ -X s ) k+1 with { L[p,k] } ∈ (sl 2 (C)) r-1 /GL 2 (C) (2-1)
where r = r ∞ + n s=1 r s and GL 2 (C) acts simultaneously by conjugation on all coefficients

{ L[p,k] } p,k .
Let us denote FR,r the subspace in F R,r composed of elements with coefficients { L[∞,k] } 0≤k≤r∞-1 ∪ { L[Xs,k] } 1≤s≤n,1≤k≤rs-1 having distinct eigenvalues.

Remark 2.1. In the present article, we shall assume that ∞ is always a pole following the standard convention. Of course, one may always use a change of coordinates in order to remove such assumption. F R,r can be given a Poisson structure inherited from the Poisson structure of a corresponding loop algebra [START_REF] Harnad | Dual isomonodromic deformations and moment maps to loop algebras[END_REF][START_REF] Adams | Dual moment maps into loop algebras[END_REF][START_REF] Woodhouse | Duality for the general isomonodromy problem[END_REF]. It is a Poisson space of dimension dim F R,r = 4r -7.

(2-2)

The space F R,r has been intensively studied from the point of view of isospectral and isomonodromic deformations. Following P. Boalch's works [START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF], one can use the Poisson structure on F R,r in order to describe it as a bundle whose fibers are symplectic leaves obtained by fixing the irregular type and monodromies of L(λ). Let us briefly review this perspective and use it to define local coordinates on F R,r trivializing the fibration.

In this article, we shall restrict to FR,r to simplify the presentation but the present setup may surely be extended to F R,r .

For any pole p ∈ R, let us define a local coordinate

∀ p ∈ R , z p (λ) := (λ -p) if p ∈ {X 1 , . . . , X n } λ -1 if p = ∞ (2-3)
Given L(λ) in an orbit of FR,r and a pole p ∈ R, there exists a gauge matrix G p ∈ GL 2 [[z p (λ)]] holomorphic at λ = p such that the gauge transformation Ψ p = G p Ψ provides

Ψp(λ) = Ψ (reg) p (λ) diag   exp   - rp-1 k=1 t p,k kzp(λ) k + tp,0 ln zp(λ)   , exp   rp-1 k=1 t p,k kzp(λ) k -tp,0 ln zp(λ)     (2-4)
where Ψ

(reg) p (λ) is regular at λ = p. It corresponds to a Lax matrix

L p = G p L(λ)G -1 p + (∂ λ G p )G -1 p satisfying G p L(λ)dλ G -1 p + (∂ λ G p )G -1 p dλ = dD p (z p ) -Λ p dz p z p where D p (z p ) = rp-1 k=1 Q p,k z k p (2-5)
with D p,k = diag -t p,k k , t p,k k and Λ p = diag(-t p,0 , t p,0 ) , ∀ p ∈ R [START_REF] Adams | Darboux coordinates on coadjoint orbits of Lie algebras[END_REF][START_REF] Adams | Dual moment maps into loop algebras[END_REF][START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF] for some complex numbers (t p,k ) p∈R,0≤k≤rp-1 . D p (z p ) is called the irregular type of L at p and Λ p its residue (also called monodromy). Equation [START_REF] Adams | Darboux coordinates on coadjoint orbits of Lie algebras[END_REF][START_REF] Adams | Dual moment maps into loop algebras[END_REF][START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF] is known as the Birkhoff factorization or formal normal solution or Turritin-Levelt fundamental form. We shall denote t = (t p,k ) p∈R,1≤k≤rp-1 the irregular times while the residues t 0 := (t p,0 ) p∈R will be referred to as monodromies by abuse of language.

Remark 2.2. In the literature, the set of irregular times t = (t p,k ) p∈R,1≤k≤rp-1 is referred to as "spectral times" or "KP times". This terminology originates from the study of isospectral systems and do not include the monodromy parameters (t p,0 ) p∈R .

Representative normalized at infinity

Fixing the irregular times t and monodromies t 0 of L(λ) does not fix it uniquely. In each orbit in FR,r , there exists a unique element such that L[∞,r∞-1] is diagonal and such that L[∞,r∞-2] takes the form

Res λ→∞ L(λ)λ -(r∞-2) = - β r∞-2 ω δ r∞-2 -β r∞-2 . (2-7)
where ω is a given non-zero constant. 1In particular, one may identify FR,r with the space of such representatives:

• If r ∞ ≥ 3: FR,r := L(λ) = r∞-1 k=1 L[∞,k] λ k-1 + n s=1 rs-1 k=0 L[Xs,k] (λ-Xs) k+1 such that { L[∞,k] } 1≤k≤r∞-1 ∪ { L[Xs,k] } 1≤s≤n,0≤k≤rs-1 ∈ (sl 2 (C)) r-1 have distinct eigenvalues, L[∞,r∞-1] = diag(-t ∞,r∞-1 , t ∞,r∞-1 ) and L[∞,r∞-2] = β r∞-2 ω δ r∞-2 -β r∞-2
, (β r∞-2 , δ r∞-2 ) ∈ C 2

(2-8)

• If r ∞ = 2: FR,r := L(λ) = L[∞,1] + n s=1 rs-1 k=0 L[Xs,k] (λ-Xs) k+1 such that { L[∞,1] } ∪ { L[Xs,k] } 1≤s≤n,0≤k≤rs-1 ∈ (sl 2 (C)) r-1 have distinct eigenvalues, L[∞,1] = diag(-t ∞,1 , t ∞,1 )
and

n s=1 L[Xs,0] = β 0 ω δ 0 -β 0 , (β 0 , δ 0 ) ∈ C 2 (2-9) • If r ∞ = 1: FR,r := L(λ) = n s=1 rs-1 k=0 L[Xs,k] (λ-Xs) k+1 such that { L[Xs,k] } 1≤s≤n,0≤k≤rs-1 ∈ (sl 2 (C)) r-1 have distinct eigenvalues, n s=1 L[Xs,0] = diag(-t ∞,0 , t ∞,0 ) and n s=1 L[Xs,1] + n s=1 X s L[Xs,0] = β -1 ω δ -1 -β -1 , (β -1 , δ -1 ) ∈ C 2 (2-10)
In the following, we shall use the notation L(λ) whenever we consider such a representative and we shall call it a representative "normalized at infinity" to stress that the GL 2 (C) global action has been used to select a specific form at infinity. Note that the choice of a representative is necessary to have uniquely defined Lax matrices. Of course, one may always perform a gauge transformation to select another type of normalization.

Remark 2.3. The Hamiltonian system is independent of the choice of representative. In the isospectral approach, the part of the GL 2 (C) action corresponding to the action of diagonal matrices (that has been used here to fix the subleading order at infinity with a ω in the upperright entry) may be translated at the level of coordinates. In [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF], this corresponds to the reduction of the (x, y) variables to the (u, v) variables.

Remark 2.4. The choice of normalization at infinity, implies that coefficient β r∞-2 identifies with -t ∞,r∞-2 for r ∞ ≥ 2. Indeed, the diagonalization of the singular part given by (2-5) implies that

det( L(λ) + G -1 ∞ ∂ λ G ∞ ) λ→∞ = - r∞-1 k=0 t ∞,k λ k-1 + O(λ -2 ) 2 = -t 2 ∞,r∞-1 λ 2r∞-4 -2t ∞,r∞-1 t ∞,r∞-2 λ 2r∞-5 + O(λ 2r∞-6 ) (2-11)
The l.h.s. is of the form:

• If r ∞ ≥ 3: det( L(λ) + G -1 ∞ ∂ λ G∞) λ→∞ = -t∞,r ∞-1 λ r∞-2 + βr ∞ -2λ r∞-3 + O(λ r∞-4 ) ωλ r∞-3 + O(λ r∞-4 ) δr ∞-2 λ r∞-3 + O(λ r∞-4 ) t∞,r ∞ -1λ r∞-2 -βr ∞-2 λ r∞-3 + O(λ r∞-4 ) = -t 2 ∞,r∞-1 λ 2r∞-4 + 2βr ∞-2 t∞,r ∞-1 λ 2r∞-5 + O(λ 2r∞-6 ) (2-12)
Identifying with the r.h.s. of [START_REF] Adams | Darboux coordinates on coadjoint orbits of Lie algebras[END_REF][START_REF] Adams | Dual moment maps into loop algebras[END_REF][START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF] we get

β r∞-2 = -t ∞,r∞-2 . • If r ∞ = 2, we observe that the matrix G ∞ (λ) is of the form G ∞ (λ) = G 0 +G 1 λ -1 +O(λ -2 )
with G 0 diagonal (in order to preserve the fact that the leading order is already diagonal).

Consequently

G -1 ∞ ∂ λ G ∞ = O(λ -2 ). The diagonalization (2-5) implies: det( L(λ) + G -1 ∞ ∂ λ G ∞ ) λ→∞ = -t 2 ∞,1 -2t ∞,1 t ∞,0 λ -1 + O(λ -2 ) (2-13) Since G -1 ∞ ∂ λ G ∞ = O(λ -2 ), the l.h.s. is of the form det( L(λ) + G -1 ∞ ∂ λ G ∞ ) λ→∞ = -t ∞,1 + β 0 λ -1 + O(λ -2 ) ωλ -1 + O(λ -2 ) δ 0 λ -1 + O(λ -2 ) t ∞,1 -β 0 λ -1 + O(λ -2 ) = -t 2 ∞,1 + 2β 0 t ∞,1 λ -1 + O(λ -2 ) (2-14)
Identifying with the r.h.s. of (2-13) we get β 0 = -t ∞,0 .

• If r ∞ = 1 we simply have

β -1 = n s=1 L[Xs,1] + X s L[Xs,0] 1,1 (2-15)

General isomonodromic deformations

The irregular times t and monodromies t 0 provide a natural set of parameters parametrizing FR,r . As the terminology suggests, a general isomonodromic deformation corresponds to a differential operator of the form.

Definition 2.2. We define the following general deformation operators.

L α = r∞-1 k=1 α ∞,k ∂ t ∞,k + n s=1 rs-1 k=1 α Xs,k ∂ t Xs,k + n s=1 α Xs ∂ Xs (2-16)
where we define the vector α ∈ C g+2-n by

α = r∞-1 k=1 α ∞,k e ∞,k + n s=1 rs-1 k=1 α Xs,k e Xs,k + n s=1
α Xs e Xs .

(2-17)

Note that the vector α may depend on both the irregular times and the monodromies. Associated to a general isomonodromic deformation is a matrix Ãα (λ, t, t 0 ) such that

L α [ Ψ(λ, t, t 0 )] = Ãα (λ, t, t 0 ) Ψ(λ, t, t 0 ) (2-18)
The important point is that Ãα (λ, t, t 0 ) is meromorphic in λ with a pole structure dominated by those ot L. The compatibility of

∂ λ Ψ(λ, t, t 0 ) = L(λ, t, t 0 ) Ψ(λ, t, t 0 ) L α [ Ψ(λ, t, t 0 )] = Ãα (λ, t, t 0 ) Ψ(λ, t, t 0 ) (2-19)
is equivalent to the isomonodromic compatibility equations

L α [ L(λ, t, t 0 )] -∂ λ Ãα (λ, t, t 0 ) + L(λ, t, t 0 ), Ãα (λ, t, t 0 ) = 0 (2-20)
which corresponds to a zero curvature equation for connections.

In [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF], the authors showed that one may define g isomonodromic times (τ i ) 1≤i≤g from the irregular times along with some additional trivial times T 1 and T 2 . It is then proven that the Hamiltonian structure contains no dependence on the trivial times so that the Darboux coordinates (q i , p i ) 1≤i≤g depend uniquely on the isomonodromic times. The existence of two trivial times in the sl 2 (C) case is directly related to the action of the Möbius transformations on the Lax matrix. Indeed, it is known that the Hamiltonian system is invariant under the Möbius transformation while the Lax matrix remains dependent on this choice. Since we have assumed that infinity is always a pole, the Möbius transformations reduce to only translations and dilations and hence has two independent generators giving rise to two trivial times. The normalization of the Lax matrix at infinity suggests some specific values for the trivial times hence implying some specific values for the irregular times in order to obtain compact results. In [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF], such convenient choices were discussed and depend on the degree of the pole at infinity. We shall make the same choice in the rest of the paper that we list here. Proposition 2.1 (Definition of trivial and isomonodromic times [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF]). We have:

• If r ∞ ≥ 3, then we shall take t ∞,r∞-1 = 1 and t ∞,r∞-2 = 0. The isomonodromic times are defined by:

τ ∞,k = t ∞,k , ∀ k ∈ 1, r ∞ -3 τ Xs,k = t Xs,k , ∀ (s, k) ∈ 1, n × 1, r s -1 Xs = X s , ∀ s ∈ 1, n (2-21)
• If r ∞ = 2, then we shall take t ∞,1 = 1 and X 1 = 0. The isomonodromic times are defined by:

τ Xs,k = t Xs,k , ∀ (s, k) ∈ 1, n × 1, r s -1 Xs = X s , ∀ s ∈ 2, n (2-22)
• If r ∞ = 1 and n ≥ 2, then we shall take X 1 = 0 and X 2 = 1. The isomonodromic times are defined by:

τ Xs,k = t Xs,k , ∀ (s, k) ∈ 1, n × 1, r s -1 Xs = X s , ∀ s ∈ 2, n (2-23)
• If r ∞ = 1 and n = 1, then we shall take X 1 = 0 and t X 1 ,r 1 -1 = 1. The isomonodromic times are defined by:

τ X 1 ,k = t X 1 ,k , ∀ k ∈ 1, r 1 -2 (2-24)
Remark 2.5. Note that in [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF], the isomonodromic times corresponding to the irregular times were defined with a factor two (i.e. τ p,k = 2t p,k ) that we omit in this paper to have simpler relations with isospectral deformations. This simply corresponds to a trivial rescaling of the corresponding isomonodromic times. However, one needs to adapt these factors when matching the current results with the expressions obtained in [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF].

Remark 2.6. For compactness and to match with results of [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF], we may write some formulas keeping all irregular times to arbitrary values but one needs to remember that the upcoming formulas are only valid when the values of t ∞,r∞-1 , t ∞,r∞-2 , t X 1 ,r 1 -1 , X 1 and X 2 are exactly the values specified in Proposition 2.1 depending on the values of r ∞ and n.

3 Lax pairs and Hamiltonians in the oper gauge

This section presents a summary of results of [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF] that shall be used in the present article. The main purpose of [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF] was to obtain explicit formulas for the Lax matrices and the Hamiltonians in the oper gauge where the Lax matrix is a companion matrix. In this gauge, the natural Darboux coordinates are (q, p) := {q 1 , . . . , q g , p 1 , . . . , p g }. We refer to [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF] for details on the derivations of the results presented in this section.

Starting from the Lax system ∂ λ Ψ(λ) = L(λ) Ψ(λ) normalized at infinity according to Section 2.2, the first step of [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF] is to perform a gauge transformation to switch the setting to the oper form (i.e. a companion-like form for the Lax matrix) of the differential system. This step is also equivalent to write the so-called "quantum curve" satisfied by Ψ1,1 (λ) and Ψ1,2 (λ). Proposition 3.1 (Gauge transformation to oper form [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF]). Let us define S ∞ := 0, r ∞ -4 and S Xs := 1, r s , ∀ s ∈ 1, n and

G(λ) := 1 0 L1,1 (λ) L1,2 (λ) =     1 0 -Q(λ)-(t ∞,r∞-1 λ+g 0 ) g j=1 (λ-q j ) n s=1 (λ-Xs) rs ω g j=1 (λ-q j ) n s=1 (λ-Xs) rs     (3-1)
where g := r ∞ -3 + n s=1 r s and

Q(λ) := - g i=1 p i n s=1 (q i -X s ) rs j̸ =i λ -q j q i -q j (3-2)
and

g 0 := t ∞,r∞-2 + t ∞,r∞-1   g j=1 q j - n s=1 r s X s   if r ∞ ≥ 2 g 0 := t ∞ (1) ,0   g j=1 q j - n s=1 r s X s   - n s=1 X s L[Xs,0] + L[Xs,1] 1,1 if r ∞ = 1 (3-3)
The gauge transformation Ψ(λ) := G(λ) Ψ(λ) implies that Ψ(λ) satisfies the companion-like system (also named oper form):

∂ λ Ψ(λ) = L(λ)Ψ(λ) , L(λ) = 0 1 L 2,1 (λ) L 2,2 (λ) (3-4) with L 2,1 (λ) = -P2 (λ) + r∞-4 j=0 H ∞,j λ j + n s=1 rs j=1 H Xs,j (λ -X s ) -j - g j=1 p j λ -q j -t ∞ (1) ,r∞-1 λ r∞-3 δ r∞≥3 L 2,2 (λ) = g j=1 1 λ -q j - n s=1 r s λ -X s (3-5)
where

P2 (λ) := 2r∞-4 j=max(0,r∞-3) P (2) ∞,j λ j + n s=1 2rs j=rs+1 P (2) Xs,j (λ -X s ) j (3-6)
is defined by

P (2) ∞,2r∞-4-k := - k j=0 t ∞,r∞-1-j t ∞,r∞-1-(k-j) , ∀ k ∈ 0, r ∞ -1 , P (2)
Xs,2rs-k := -

k j=0 t Xs,rs-1-j t Xs,rs-1-(k-j) , ∀ s ∈ 1, n , ∀ k ∈ 0, r s -1 (3-7)
Moreover for r ∞ = 2, we have the additional relations

P (2) ∞,0 = -(t ∞,1 ) 2 , n s=1 H Xs,1 = g j=1 p j + (2t ∞,1 t ∞,0 -t ∞,1 ) (3-8)
while, for r ∞ = 1, we have

n s=1 H Xs,1 = g j=1 p j , n s=1 X s H Xs,1 + n s=1 H Xs,2 δ rs≥2 = g j=1 q j p j + n s=1 (t Xs,0 ) 2 δ rs=1 + t ∞,0 (t ∞,0 -1). (3-9)
Note that in Proposition 3.1, the coordinates q are defined as the apparent singularities of the system (i.e. zeros of det( Ψ(λ))). The dual coordinates p are chosen so that for any i ∈ 1, g , (q i , p i ) is a point on the spectral curve, i.e. det(p i I 2 -L(q i )) = 0 for all i ∈ 1, g . The oper gauge is particularly convenient for computations since all the information on the Hamiltonian system is contained into L 2,1 (λ), i.e. in the g unknown coefficients (H p,k ) p∈R,k∈Sp . In fact, the main result of [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF] is to prove that the coordinates (q, p) satisfy some Hamiltonian evolutions relatively to the isomonodromic times defined in Proposition 2.1 and that the Hamiltonians are easily expressed in terms of H := (H p,k ) p∈R,k∈Sp . Theorem 3.1 (Hamiltonian evolution of (q, p) (Theorem 7.1 of [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF])). For any isomonodromic time τ and the choice of irregular times defined in Proposition 2.1, we have

∂ τ q i = ∂Ham (ατ ) (q, p, t, t 0 ) ∂p i , ∂ τ p i = - ∂Ham (ατ ) (q, p, t, t 0 ) ∂q i , ∀ i ∈ 1, g (3-10)
where the Hamiltonians Ham (ατ ) (q, p, t, t 0 ) are given by:

  
Ham (αt ∞,1 ) (q, p, t, t 0 ) . . .

(r ∞ -3)Ham (αt ∞,r∞-3 ) (q, p, t, t 0 )    = (M ∞ (t)) -1   
H ∞,r∞-4 (q, p, t, t 0 ) . . .

H ∞,0 (q, p, t, t 0 )      
Ham (αt Xs,1 ) (q, p, t, t 0 ) . . .

(r s -1)Ham (αt Xs,rs-1 ) (q, p, t, t 0 )

   = (M Xs (t)) -1   
H Xs,rs (q, p) . . .

H Xs,2 (q, p)    , ∀ s ∈ 1, n   
Ham (α X 1 ) (q, p, t, t 0 ) . . .

Ham (α Xn ) (q, p, t, t 0 )    =    H X 1 ,1 (q, p, t, t 0 ) . . . H Xn,1 (q, p, t, t 0 )    (3-11)
and the matrices M ∞ (t) and (M Xs (t)) 1≤s≤n are defined by

M ∞ (t) =                t ∞,r∞-1 0 . . . . . . 0 t ∞,r∞-2 t ∞,r∞-1 0 . . . t ∞,r∞-3 t ∞,r∞-2 t ∞,
0 t ∞,3 t ∞,4 . . . t ∞,r∞-3 t ∞,r∞-2 t ∞,r∞-1                ∈ M r∞-3 (C) (3-12) and M Xs (t) =             t Xs,
            , ∀ s ∈ 1, n (3-13)
The previous theorem is supplemented by the explicit expression of the coefficients H(q, p, t, t 0 ). Proposition 3.2 (Explicit expression of the H(q, p, t, t 0 ) (Proposition 5.1 of [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF])). We have

(V∞) t (VX 1 ) t . . . (VX n ) t      H∞ HX 1 . . . HX n      =            p 2 1 + p1 n s=1 rs q 1 -Xs + P2(q1) + i̸ =1 p i -p 1 q 1 -q i + t ∞ (1) ,r∞-1 q r∞-3 1 δ r∞≥3 . . . . . . p 2 g + pg n s=1 rs qg -Xs + P2(qg) + i̸ =g p i -pg qg -q i + t ∞ (1) ,r∞-1 q r∞-3 g δ r∞≥3            (3-14)
with

H ∞ = (H ∞,0 , . . . H ∞,r∞-4 ) t (null is r ∞ ≤ 3),
q r∞-4 1 q r∞-4 2 . . . . . . q r∞-4 g          , VX s =          1 q 1 -Xs . . . . . . 1 qg -Xs 1 (q 1 -Xs) 2 . . . . . . 1 (qg -Xs) 2 . . . . . . . . . . . . 1 (q 1 -Xs) rs . . . . . . 1 (qg -Xs) rs          (3-15)
Moreover, for r ∞ = 2 (resp. r ∞ = 1) the previous linear system has to be supplemented with the additional relations (3-8) (resp. [START_REF] Adams | Dual moment maps into loop algebras[END_REF][START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF]). Finally for r ∞ = 1 we have:

g 0 = 1 2t ∞ (1) ,0 - n s=1 (2X s P (2)
Xs,2 δ rs=1 + P

(2)

Xs,3 δ rs=2 ) + n s=1 (X 2 s H Xs,1 + 2X s H Xs,2 δ rs≥2 + H Xs,3 δ rs≥3 ) - g j=1 p j q 2 j +t ∞ (1) ,0 (2t ∞ (1) ,0 -1)   g j=1 q j - n s=1 r s X s   (3-16)
Theorem 3.1 and Proposition 3.2 provide the explicit expression of the Hamiltonian system satisfied by (q, p). This Hamiltonian system is equivalent to the fact that the differential system ∂ λ Ψ(λ) = L(λ)Ψ(λ) is supplemented by the general auxiliary system

L α [Ψ(λ)] = A α (λ)Ψ(λ) (3-17)
where A α (λ) is rational in λ with poles in R dominated by those of L and simple poles at the apparent singularities (q i ) 1≤i≤g . The compatibility of the Lax system (3-4) and [START_REF] Adams | Dual moment maps into loop algebras[END_REF][START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF][START_REF] Chernyakov | Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems[END_REF][START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Fuchs | Sur quelques équations différentielles linéaires du second ordre[END_REF][START_REF] Gaiur | Isomonodromic deformations: Confluence, reduction & quantisation[END_REF][START_REF] Gambier | Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes[END_REF][START_REF] Garnier | Solution du problème de Riemann pour les systèmes différentiels linéaires du second ordre[END_REF] is equivalent to the isomonodromic compatibility equation:

L α [L(λ)] = ∂ λ A α (λ) -[L(λ), A α (λ)] (3-18)
In [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF], it is proved that the first line of A α (λ) is determined by

[A α (λ)] 1,1 = c ∞,0 - g j=1 p j µ (α) j λ -q j [A α (λ)] 1,2 = ν (α) ∞,-1 λδ r∞=1 + ν (α) ∞,0 δ r∞≤2 + g j=1 µ (α) j λ -q j (3-19) with 2 c ∞,0 = 1 2ω L α [ω] + 1 2 ν (α) ∞,-1 δ r∞=1 (3-20)
while the second line of A α (λ) is immediately determined by the first line of the isomonodromic compatibility equation (3-18) (which is always trivial in the oper gauge):

[A α (λ)] 2,1 = ∂ λ [A α (λ)] 1,1 + [A α (λ)] 1,2 L 2,1 (λ), [A α (λ)] 2,2 = ∂ λ [A α (λ)] 1,2 + [A α (λ)] 1,1 + [A α (λ)] 1,2 L 2,2 (λ), (3-21)
Coefficients µ

(α) j 1≤j≤g
are determined by

        V ∞ V 1 . . . . . . V n                µ (α) 1 . . . . . . µ (α) g        =       ν (α) ∞ -ν (α) X1 . . . -ν (α) Xn       with ν (α) ∞ =       ν (α) ∞,1 ν (α) ∞,2 . . . ν (α) ∞,r∞-3       , ν (α)
Xs =       ν (α) Xs,0 -X s ν (α) ∞,-1 δ r∞=1 -ν (α) ∞,0 δ r∞≤2 ν (α) Xs,1 -ν (α) ∞,-1 δ r∞=1 . . . ν (α) Xs,rs-1       (3-22) Coefficients ν (α) p,k p,k
correspond to the expansion of [A α (λ)] 1,2 at each pole. Under conditions of Proposition 2.1, we have:

[A α (λ)] 1,2 λ→∞ = ν (α) ∞,-1 δ r∞=1 λ + ν (α) ∞,0 δ r∞≤2 + r∞-3 i=1 ν (α) ∞,i λ i + O λ -(r∞-2) , [A α (λ)] 1,2 λ→Xs = rs-1 i=0 ν (α) Xs,i (λ -X s ) i + O ((λ -X s ) rs ) (3-23)
A crucial technical point is given by the following proposition.

Y (λ) := - r∞-1 k=0 t ∞,k λ k-1 + n s=1 rs-1 k=0 t Xs,k (λ -X s ) k+1 a(λ) := - r∞-3 k=1 α ∞,k k λ k - n s=1 rs-1 k=1 α Xs,k (λ -X s ) k - n s=1 rs-1 k=0 α Xs t Xs,k (λ -X s ) k+1 (3-24)
then, coefficients ν

(α) p,k p,k
are determined by

Y (λ) [A α (λ)] 1,2 λ→∞ = a(λ) + O (1) , Y (λ) [A α (λ)] 1,2 λ→Xs = a(λ) + O (1) (3-25)
which is equivalent to (see Appendix E of [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF] for details)

ν (α) Xs,0 = -α Xs , M Xs     ν (α) Xs,1 . . . ν (α) Xs,rs-1     = -    α Xs,rs-1 rs-1 . . . α Xs,1 1    , M ∞     ν (α) ∞,1 . . . ν (α) ∞,r∞-3     =    α ∞,r∞-3 r∞-3 . . . α ∞,1 1    (3-26)
Let us finally make some comments about the advantages and drawbacks of the previous formulas.

• The oper gauge and the associated Darboux coordinates (q, p) allow the explicit resolution of the compatibility equations [START_REF] Adams | Dual moment maps into loop algebras[END_REF][START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF][START_REF] Chernyakov | Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems[END_REF][START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Fuchs | Sur quelques équations différentielles linéaires du second ordre[END_REF][START_REF] Gaiur | Isomonodromic deformations: Confluence, reduction & quantisation[END_REF][START_REF] Gambier | Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes[END_REF][START_REF] Garnier | Solution du problème de Riemann pour les systèmes différentiels linéaires du second ordre[END_REF][START_REF] Harnad | Dual isomonodromic deformations and moment maps to loop algebras[END_REF] and the explicit expression of the Hamiltonian system via Theorem 3.1 and Proposition 3.2.

• The relation between the Hamiltonians and the spectral invariants H of L(λ) is very simple (eq. [START_REF] Adams | Dual moment maps into loop algebras[END_REF][START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF]) and recovers results of [START_REF] Gaiur | Isomonodromic deformations: Confluence, reduction & quantisation[END_REF] obtained by confluences of simple poles.

• The Hamiltonian system has a complicated dependence in the Darboux coordinates (q, p).

In particular, it has singularities at q i = q j for i ̸ = j.

• One may obtain the expression of the geometric Lax matrices L(λ) and Ã(λ) in terms of (q, p) using the explicit gauge transformation (3-1):

L(λ) = G(λ) -1 L(λ)G(λ) -G(λ) -1 ∂ λ G(λ) Ãα (λ) = G(λ) -1 A α (λ)G(λ) -G(λ) -1 L α [G(λ)] (3-27)
However, the expressions are particularly complicated and it is not obvious in the formulas that L(λ) and Ãα (λ) do not have poles at λ ∈ {q 1 , . . . , q g }.

The conclusion of this section is that the Darboux coordinates (q, p) are well-adapted to the oper gauge (L(λ), A α (λ)). The Hamiltonian in these coordinates has also a natural interpretation in terms of interacting particles. However, regarding the initial geometric Lax matrix L(λ) and the geometric setup of Section 2, coordinates (q, p) are cumbersome. Thus, the first goal of the present paper is to define some more appropriate Darboux coordinates (Q, P) to obtain explicit formulas for the Lax matrix L(λ) and explicit formulas for the Hamiltonians associated to these Darboux coordinates.

4 Geometric Darboux coordinates and associated Hamiltonian system

Geometric Darboux coordinates

As explained above, we shall introduce a new set of Darboux coordinates (Q, P) that is more adapted to the Lax matrix L(λ). We shall name these coordinates "Geometric Darboux coordinates" to stress that they are naturally adapted to the initial gauge and thus to the geometric construction of meromorphic connections on gl 2 (C).

Definition 4.1 (Geometric Darboux coordinates). We define the "geometric Darboux coordinates" (Q, P) = (Q p,k , P p,k ) p∈R,k∈Sp by

ω g j=1 (λ -q j ) n s=1 (λ -X s ) rs = n s=1 rs k=1 Q Xs,k (λ -X s ) k + r∞-4 k=0 Q ∞,k λ k + ω δ r∞≥3 λ r∞-3 p i = r∞-4 k=0 P ∞,k ∂Q ∞,k (q 1 , . . . , q g ) ∂q i + n s=1 rs k=1 P Xs,k ∂Q Xs,k (q 1 , . . . , q g ) ∂q i , ∀ i ∈ 1, g (4-1)
with the additional relations

• For r ∞ = 2: n s=1 Q Xs,1 = ω and rs m=1 P Xs,m Q Xs,m = 0 (4-2)
• For r ∞ = 1: Conditions (4-2) and (4-3) are derived for completeness in Lemma A.2 of Appendix A. Lemma 6.3 of [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlevé 1 hierarchy[END_REF] proves that the change of coordinates (q, p) ↔ (Q, P) is a time-independent symplectic change of coordinates. Consequently, one may obtain the Hamiltonian in terms of (Q, P) using Theorem 3.1 and replacing the coordinates (q, p) in Proposition 3.2 in terms of (Q, P). However, this approach is very cumbersome in practice and we shall derive a more convenient formula to obtain the Hamiltonians relatively to (Q, P) in Theorem 4.2.

n s=1 Q Xs,1 = 0 and n s=1 Q Xs,2 + n s=1 X s Q Xs,1 = ω
Remark 4.1. The change of coordinates q ↔ Q is algebraically natural. It simply corresponds to writing a rational function either in its factorized form or in its polar form. It turns out that the factorized form is more convenient in the oper gauge while the polar form is more adapted to the geometric gauge. The remaining part of the change of coordinates giving p in terms of (Q, P) follows naturally to preserve the symplectic structure.

Expression of the geometric Lax matrices

Using the new coordinates of Definition 4.1, we may write the expression of the Hamiltonian and of the geometric Lax matrices L and Ãα . In order to obtain formulas, we shall introduce the following notation: Definition 4.2. Let a ∈ C. For any function f (λ) admitting a Laurent series at λ → a we shall denote [f (λ)] a,-the singular part at λ → a:

f (λ) = ∞ k=-r F k (λ -a) k ⇒ [f (λ)] a,-= r k=1 F -k (λ -a) -k (4-4)
For any function f (λ) admitting a Laurent series at λ → ∞, we shall denote [f (λ)] ∞,+ the polynomial part at infinity (including the O(1) term):

f (λ) = ∞ k=-r F k λ -k ⇒ [f (λ)] ∞,+ = r k=0 F -k λ k (4-5)
Theorem 4.1 (Geometric Lax matrices in terms of geometric Darboux coordinates). We have:

L1,2 (λ) = n s=1 rs k=1 Q Xs,k (λ -X s ) k + r∞-4 k=0 Q ∞,k λ k + ωδ r∞≥3 λ r∞-3 L1,1 (λ) = -ω r∞-4 k=0 P ∞,r∞-4-k λ k - r∞-5 k=0 r∞-5-k m=0 P ∞,m Q ∞,k+1+m λ k + n s=1 rs k=1 rs+1-k m=1 P Xs,m Q Xs,k+m-1 (λ -X s ) -k - (t ∞,r∞-1 λ + g 0 ) ω ω δ r∞≥3 λ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ -X s ) k L2,2 (λ) = -L1,1 (λ) Ãα (λ) 1,2 = ω ν (α) ∞,1 λ r∞-4 δ r∞≥4 + r∞-5 j=0   ω ν (α) ∞,r∞-3-j + r∞-4 k=j+1 ν (α) ∞,k-j Q ∞,k   λ j + n s=1 rs j=1   rs k=j ν (α) Xs,k-j Q Xs,k   (λ -X s ) -j (4-6)
and

Ãα(λ) 1,1 = 1 2ω Lα[ω] -t∞,1ν (α)
∞,0 δr ∞,2 + 1 2 -t∞,0 ν (α) ∞,-1 δr ∞,1 -t∞,r ∞-1 ν (α) ∞,1 λ r∞-3 δ r∞≥3 -t∞,r ∞-1 ν (α) ∞,2 λ r∞-4 δ r∞≥4 -t∞,r ∞ -1ν (α) ∞,3 + ωP∞,0 + t∞,r ∞-1 Q∞,r ∞-5 + Q∞,r ∞ -4g0 ω ν (α) ∞,1 λ r∞-5 δ r∞≥5 - r∞-6 j=0 t∞,r ∞-1 ν (α) ∞,r∞-2-j + ν (α) ∞,r∞-4-j ωP∞,0 + t∞,r ∞-1 Q∞,r ∞-5 + Q∞,r ∞ -4g0 ω + r∞-5-j i=1 ν (α) ∞,i ωP∞,r ∞-4-i-j + r∞-5-i-j m=0 P∞,mQ∞,j+i+1+m + t∞,r ∞-1 Q∞,j+i-1 + g0Q∞,j+i ω λ j + n s=1 PX s,1 - t∞,r ∞ -1Xs + g0 ω QX s ,rs ν (α)
Xs,0 (λ -Xs) -rs

+ n s=1 rs-1 r=1 rs-r i=0 rs+1-r-i m=1 ν (α) Xs,i PX s,m QX s ,r+i+m-1 - (t∞,r ∞ -1Xs + g0) ω rs-r i=0 ν (α) Xs,i QX s,r+i - t∞,r ∞ -1 ω rs-1-r i=0 ν (α) Xs,i QX s,r+i+1 (λ -Xs) -r Ãα(λ) 2,2 = -Ãα(λ) 1,1 (4-7)
with the extra condition for r ∞ = 2:

n s=1 rs k=1 ν (α) Xs,k-1 Q Xs,k = ω ν (α) ∞,0 (4-8)
and the extra conditions for r ∞ = 1:

ω ν (α) ∞,-1 = n s=1 rs k=1 ν (α) Xs,k-1 Q Xs,k n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 ν (α) ∞,-1 + ω ν (α) ∞,0 = n s=1 rs k=2 ν (α) Xs,k-2 Q Xs,k + X s rs k=1 ν (α) Xs,k-1 Q Xs,k (4-9)
The expression of L2,1 (λ) depends on the value of r ∞ :

• If r ∞ ≥ 3: L2,1 (λ) =       2r∞-4 j=r∞-3 2r∞-4-j m=0 t ∞,r∞-1-m t ∞,j+m-r∞+3 λ j -L1,1 (λ) 2 ωλ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k       ∞,+ + n s=1       2rs j=rs+1 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 (λ -X s ) -j -L1,1 (λ) 2 ωλ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k       Xs,- (4-10) • If r ∞ = 2: L2,1 (λ) = n s=1       2rs j=rs+1 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 (λ -X s ) -j -L1,1 (λ) 2 n s=1 rs k=1 Q Xs,k (λ-Xs) k       Xs,- (4-11) • If r ∞ = 1: L2,1 (λ) = n s=1       2rs j=rs+1 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 (λ -X s ) -j -L1,1 (λ) 2 L1,2 (λ)       Xs,- + 2(t ∞,0 λ + g 0 ) ω L1,1 (λ) - (t ∞,0 λ + g 0 ) 2 ω 2 L1,2 (λ) + (t ∞,0 ) 2 ω (4-12) with L1,1 (λ) := L1,1 (λ) + (t ∞,0 λ + g 0 ) ω L1,2 (λ) = n s=1 rs k=1 rs+1-k m=1 P Xs,m Q Xs,k+m-1 (λ -X s ) -k (4-13)
The expression of [ Ãα (λ)] 2,1 also depends on the value of r ∞ :

• For r ∞ ≥ 4: [ Ãα (λ)] 2,1 = - t ∞,r∞-1 ω 2 L α [ω]λ + 2 ω Res λ→∞ Ã(α) 1,1 (λ) - L(α) 1,1 (λ) L(α) 1,2 (λ) Ã(α) 1,2 (λ) + 1 ω 3 Q ∞,r∞-4 L α [ω] +       2r∞-5 k=r∞-3 2r∞-4 j=k+1 2r∞-4-j m=0 t ∞,r∞-1-m t ∞,j+m-r∞+3 ν (α) ∞,j-k λ k ωλ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k       ∞,+ + L1,1 (λ) L1,2 (λ) L1,1 (λ) L1,2 (λ) [ Ãα (λ)] 1,2 -2[ Ãα (λ)] 1,1 ∞,+ + n s=1       2rs k=rs+1 2rs j=k 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 ν (α) Xs,j-k (λ -X s ) -k ωλ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k       Xs,- + n s=1 L1,1 (λ) L1,2 (λ) L1,1 (λ) L1,2 (λ) [ Ãα (λ)] 1,2 -2[ Ãα (λ)] 1,1 Xs,- (4-14)
• For r ∞ = 3:

[ Ãα (λ)] 2,1 = - t ∞,2 ω 2 L α [ω]λ + 2 ω Res λ→∞ Ã(α) 1,1 (λ) - L(α) 1,1 (λ) L(α) 1,2 (λ) Ã(α) 1,2 (λ) + 1 ω 3 n s=1 Q Xs,1 L α [ω] +       2r∞-5 k=r∞-3 2r∞-4 j=k+1 2r∞-4-j m=0 t ∞,r∞-1-m t ∞,j+m-r∞+3 ν (α) ∞,j-k λ k ωλ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs ,k (λ-Xs) k       ∞,+ + L1,1 (λ) L1,2 (λ) L1,1 (λ) L1,2 (λ) [ Ãα (λ)] 1,2 -2[ Ãα (λ)] 1,1 ∞,+ + n s=1       2rs k=rs+1 2rs j=k 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 ν (α) Xs,j-k (λ -X s ) -k ωλ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k       Xs,- + n s=1 L1,1 (λ) L1,2 (λ) L1,1 (λ) L1,2 (λ) [ Ãα (λ)] 1,2 -2[ Ãα (λ)] 1,1 Xs,- (4-15)
• For r ∞ = 2:

[ Ãα (λ)] 2,1 = n s=1       2rs k=rs+1 2rs j=k 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 ν (α) Xs,j-k (λ -X s ) -k n s=1 rs k=1 Q Xs,k (λ-Xs) k       Xs,- + n s=1 L1,1 (λ) L1,2 (λ) L1,1 (λ) L1,2 (λ) [ Ãα (λ)] 1,2 -2[ Ãα (λ)] 1,1 Xs,- (4-16)
• For r ∞ = 1:

[ Ãα (λ)] 2,1 = - 2 ω 2 L α [ω] g 0 + t ∞,0 ω n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 +       2rs k=rs+1 2rs j=k 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 ν (α) Xs,j-k (λ -X s ) -k n s=1 rs k=1 Q Xs,k (λ-Xs) k       Xs,- + n s=1 L1,1 (λ) L1,2 (λ) L1,1 (λ) L1,2 (λ) [ Ãα (λ)] 1,2 -2[ Ãα (λ)] 1,1 Xs,- (4-17)
In the previous formulas, coefficients ν

(α) p,k p,k
are defined by Proposition 3.3 and ν

(α) ∞,r∞-2 is determined (only for r ∞ ≥ 3) by ω ν (α) ∞,r∞-2 = n s=1 rs k=1 ν (α) Xs,k-1 Q Xs,k - r∞-3 j=1 ν (α) ∞,j Q ∞,j-1 ω ν (α) ∞,r∞-1 = n s=1 rs k=2 ν (α) Xs,k-2 Q Xs,k + n s=1 rs k=1 ν (α) Xs,k-1 X s Q Xs,k - r∞-2 j=2 ν (α) ∞,j Q ∞,j-2 -ν (α) ∞,1 n s=1 Q Xs,1 (4-18)
Moreover, we have

g 0 = t ∞,r∞-2 - t ∞,r∞-1 ω Q ∞,r∞-4 Prop. 2.1 = - Q ∞,r∞-4 ω if r ∞ ≥ 4 g 0 = t ∞,1 - t ∞,2 ω n s=1 Q Xs,1 Prop. 2.1 = - 1 ω n s=1 Q Xs,1 if r ∞ = 3 g 0 = t ∞,0 - t ∞,1 ω n s=1 Q Xs,2 + X s Q Xs,1 Prop. 2.1 = t ∞,0 - 1 ω n s=1 Q Xs,2 + X s Q Xs,1 if r ∞ = 2 g 0 = 1 ω 1 2 -t ∞,0 n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 - 1 2t ∞,0 Res λ→∞ λ 2 ( L1,1 (λ)) 2 + ∂ λ L1,1 (λ) - t ∞,0 ω L1,2 (λ) -L1,1 (λ) ∂ λ L1,2 (λ) L1,2 (λ) + (t ∞,0 ) 2 ω L1,2 (λ) + L1,2 (λ) n s=1      2rs j=rs+1 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 (λ -X s ) -j -L1,1 (λ) 2 L1,2 (λ)      Xs,- if r ∞ = 1 (4-19)
Proof. The proof is done in Appendix A. The strategy consists in observing that the entries of the matrices L(λ) and Ãα (λ) are rational functions of λ with poles only in {∞, X 1 , . . . , X n }. Thus, rewriting the gauge transformation (3-1) using the new set of coordinates (Lemma A.2), one may obtain the behavior of each entry at each pole and finally obtain the expressions proposed in Theorem 4.1. The fact that the expressions depend on r ∞ comes from the normalization at infinity of the Lax matrices that implies to split cases in the study the local behavior at infinity of some of their entries.

Remark 4.2. For r ∞ = 1, it is convenient to introduce L1,1 (λ) to obtain a formula for g 0 that is consistent. In particular, the matrix L is independent of g 0 because geometrically the oper gauge is universal and does not depend on the choice of normalization of L. This is the reason why we had to isolate the dependence of g 0 in L2,1 (λ) for r ∞ = 1. One could obtain similar simplifications for r ∞ = 2 and r ∞ = 3 but it is less relevant since the value of g 0 is easily obtained from the knowledge of β -1 given by Remark 2.4 so that one can use it directly in the formulas.

Expression of the Hamiltonian

There are at least 3 different ways to obtain the expression of the Hamiltonian in terms of the new coordinates of Definition 4.1.

• Since the change of coordinates in Definition 4.1 is time-independent and symplectic, one may simply replace the variables (q, p) in terms of (Q, P) in the expression of the Hamiltonian. In particular, Theorem 3.1 remains valid so that one only needs to express (H p,k ) p∈R,k∈Sp in terms of the (Q, P) in Proposition 3.2.

• From Theorem 4.1, we have the explicit expression of the Lax matrices in terms of the new coordinates (Q, P). Thus, one may obtain the evolution equations of the coordinates (Q, P) by solving the compatibility equation of this system.

• One may use the gauge transformation (expressed in terms of the coordinates (Q, P) using Lemma A.2) and connect (H p,k ) p∈R,k∈Sp in terms of the expansion of det L(λ) (i.e. the spectrum of L) at each pole. Using the known expression for L(λ) given by Theorem 4.1, one may easily obtain the expression of the Hamiltonian in terms of the coordinates (Q, P).

At the computational level, the third approach seems the easiest. Indeed, the first one requires to express (q i ) 1≤i≤g in terms of (Q p,k ) p∈R,k∈Sp , that is to say to invert Definition 4.1 and plug the results into Proposition 3.2. Although this is possible in principle, it would require substantial computations and identities similar to the ones obtained in [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlevé 1 hierarchy[END_REF] for the simpler case of the Painlevé 1 hierarchy. The second option requires to solve the compatibility equations, i.e. to redo the works of [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF] using a different gauge and different coordinates. It is well-known that solving directly the compatibility equations in the geometric gauge is a difficult task and it was precisely to avoid doing so that the authors of [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF] turned themselves to the oper gauge. Consequently, we shall develop the third approach in the rest of this paper. As we will see below, this approach is also very convenient to make the connection with the isospectral deformations.

It is a straightforward computation from the gauge transformation (3-1) to prove that

-det L(λ) = L 2,1 (λ) = ( L1,1 ) 2 + L2,1 L1,2 + L1,2 ∂ λ L1,1 L1,2 (4-20)
which is also recalled in (A-71). Since the change of coordinates (q, p) → (Q, P) is timeindependent and symplectic, we immediately obtain from Theorem 3.1 the following main result.

Theorem 4.2 (Expression of the Hamiltonian in terms of (Q, P)). We have

H Xs,j (Q, P, t, t 0 ) = Res λ→Xs (λ -X s ) j-1 ( L1,1 ) 2 + L2,1 L1,2 + L1,2 ∂ λ L1,1 L1,2 , ∀ (s, j) ∈ 1, n × 1, r s H ∞,j (Q, P, t, t 0 ) = -Res λ→∞ λ -j-1 ( L1,1 ) 2 + L2,1 L1,2 + L1,2 ∂ λ L1,1 L1,2 , ∀ j ∈ 0, r ∞ -4 (4-21)
where ( L1,1 , L1,2 , L2,1 ) are given by Theorem 4.1. For any isomonodromic time τ and the choice of irregular times defined in Proposition 2.1, the Hamiltonians relatively to the coordinates (Q, P) are given by

   Ham (αt ∞,1 ) (Q, P, t, t 0 ) . . . (r ∞ -3)Ham (αt ∞,r∞-3 ) (Q, P, t, t 0 )    = (M ∞ (t)) -1    H ∞,r∞-4 (Q, P, t, t 0 ) . . . H ∞,0 (Q, P, t, t 0 )      
Ham (αt Xs,1 ) (Q, P, t, t 0 ) . . .

(r s -1)Ham (αt Xs,rs-1 ) (Q, P, t 0 )

   = (M Xs (t)) -1   
H Xs,rs (Q, P, t, t 0 ) . . .

H Xs,2 (Q, P, t, t 0 )    , ∀ s ∈ 1, n    Ham (α X 1 ) (Q, P, t, t 0 ) . . . Ham (α Xn ) (Q, P, t, t 0 )    =    H X 1 ,1 (Q, P, t, t 0 ) . . . H Xn,1 (Q, P, t, t 0 )    (4-22)
with the matrices M ∞ (t) and (M Xs (t)) 1≤s≤n defined by (3-12) and (3-13).

Remark 4.3. The matrix L is independent of the choice of normalization of L because the oper gauge is geometrically independent of this choice. Consequently, H is independent of this choice too. In particular, for r ∞ = 1, it may be convenient to use L1,1 (λ) instead of L1,1 (λ) in order to avoid the use of g 0 that is more involved. The computation is detailed in (A-72) and we have the alternative expression

L 2,1 (λ) = ( L1,1 (λ)) 2 + ∂ λ L1,1 (λ) -t ∞,0 L1,2 (λ) -L1,1 (λ) ∂ λ L1,2 (λ) L1,2 (λ) + t ∞,0 L1,2 (λ) + L2,1 (λ) n s=1       2rs j=rs+1 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 (λ -X s ) -j -L1,1 (λ) 2 L1,2 (λ)       Xs,- (4-23)

Geometric Lax coordinates and associated expressions for the geometric Lax matrices

The previous change of coordinates (q, p) → (Q, P) is interesting for the Hamiltonian system because it is a time-independent symplectic change of coordinates so that one may simply replace coordinates into the Hamiltonians. However, the price to pay is that entry L1,1 (λ) involves a complicated mix of variables Q and P. In preparation for the upcoming link with isospectral deformations, it is interesting to look for another set of coordinates (Q, R) in which L1,1 (λ) would be expressed in a simpler way. This leads to the following definition.

R Xs,k = rs+1-k m=1 P Xs,m Q Xs,k+m-1 - (g 0 + t ∞,r∞-1 X s ) ω Q Xs,k - t ∞,r∞-1 ω Q Xs,k+1 δ k≤rs-1 , ∀ (s, k) ∈ 1, n × 1, r s R ∞,r∞-4 = -ω P ∞,0 - g 0 ω Q ∞,r∞-4 - t ∞,r∞-1 ω Q ∞,r∞-5 if r ∞ ≥ 4 R ∞,k = -ω P ∞,r∞-4-k - r∞-5-k m=0 P ∞,m Q ∞,k+1+m - t ∞,r∞-1 ω Q ∞,k-1 - g 0 ω Q ∞,k , ∀ k ∈ 1, r ∞ -5 R ∞,0 = -ω P ∞,r∞-4 - r∞-5 m=0 P ∞,m Q ∞,m+1 - g 0 ω Q ∞,0 - t ∞,r∞-1 ω n s=1 Q Xs,1 (4-24)
with the additional relations

n s=1 R Xs,1 = -t ∞,0 if r ∞ ≤ 2 n s=1 X s R Xs,1 + R Xs,2 δ rs≥2 = -g 0 - t ∞,0 ω n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 = β -1 if r ∞ = 1 (4-25)
We shall name the coordinates (Q, R) the "geometric Lax coordinates" since as proved below in Lemma 4.1, they are particularly convenient to express the geometric Lax matrix L(λ).

Note that the additional constraints (4-25) are direct consequences of the additional constraints for the coordinates Q in (4-2) and of the value of g 0 provided by [START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF][START_REF] Chernyakov | Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems[END_REF][START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Fuchs | Sur quelques équations différentielles linéaires du second ordre[END_REF][START_REF] Gaiur | Isomonodromic deformations: Confluence, reduction & quantisation[END_REF][START_REF] Gambier | Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes[END_REF][START_REF] Garnier | Solution du problème de Riemann pour les systèmes différentiels linéaires du second ordre[END_REF][START_REF] Harnad | Dual isomonodromic deformations and moment maps to loop algebras[END_REF][START_REF] Harnad | Hamiltonian Dynamics, Classical R-matrices and Isomonodromic Deformations[END_REF] for r ∞ ∈ {1, 2}. The previous coordinates are well-suited to express L1,1 (λ) because of the following lemma. Lemma 4.1. We have

L1,1 (λ) = n s=1 rs k=1 R Xs,k (λ -X s ) k -t ∞,r∞-1 δ r∞≥2 λ r∞-2 -t ∞,r∞-2 δ r∞≥3 λ r∞-3 + r∞-4 k=0 R ∞,k λ k (4-26)
Proof. The proof is obvious from the expression of L1,1 (λ) given by Theorem 4.1 but is detailed for completeness in Appendix B.

For completeness, we shall mention that the inverse change of coordinates amounts to

     R ∞,r∞-4 . . . R ∞,1 R ∞,0      = -      ω 0 . . . 0 Q ∞,r∞-4 ω 0 . . . . . . . . . . . . Q ∞,1 . . . Q ∞,r∞-4 ω           P ∞,0 . . . P ∞,r∞-5 P ∞,r∞-4      - g 0 ω      Q ∞,r∞-4 . . . Q ∞,1 Q ∞,0      - t ∞,r∞-1 ω        Q ∞,r∞-5 . . . Q ∞,0 n s=1 Q Xs,1        (4-27
) and for all s ∈ 1, n :

     R Xs,rs R Xs,rs-1 . . . R Xs,1      =             Q Xs,rs 0 . . . . . . 0 Q Xs,rs-1 Q Xs,
0 Q Xs,1 Q Xs,2 . . . Q Xs,rs-1 Q Xs,rs                  P Xs,1 P Xs,2 . . . P Xs,rs      - (g 0 + t ∞,r∞-1 X s ) ω      Q Xs,rs Q Xs,rs-1 . . . Q Xs,1      - t ∞,r∞-1 ω      0 Q Xs,rs . . . Q Xs,2      (4-28)
Remark 4.4. Let us remark that the change of coordinates (Q, P) ↔ (Q, R) is time-independent but not symplectic. Therefore, one may not obtain the Hamiltonians for the coordinates (Q, R) by just replacing the coordinates (Q, P) in terms of the coordinates (Q, R) in the Hamiltonians of Theorem 4.2.

As we will see in Section 5, the coordinates (Q, R) are particularly convenient to make the connections with isospectral deformations. For completeness, we also provide the expression of the geometric Lax matrices in terms of (Q, R) in the following theorem. Theorem 4.3 (Expression of the geometric Lax matrices in terms of (Q, R)). We have:

L1,1 (λ) = n s=1 rs k=1 R Xs,k (λ -X s ) k -t ∞,r∞-1 δ r∞≥2 λ r∞-2 -t ∞,r∞-2 δ r∞≥3 λ r∞-3 + r∞-4 k=0 R ∞,k λ k L1,2 (λ) = n s=1 rs k=1 Q Xs,k (λ -X s ) k + r∞-4 k=0 Q ∞,k λ k + ωδ r∞≥3 λ r∞-3 L2,2 (λ) = -L1,1 (λ) (4-29)
Expression of L2,1 (λ) depends on the value of r ∞ and is given by (4-10) for r ∞ ≥ 3, (4-11) for r ∞ ≥ 2. For r ∞ = 1, we have

L2,1 (λ) = t ∞,0 ω - (t ∞,0 ) 2 ω + n s=1       2rs j=rs+1 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 (λ -X s ) -j -L1,1 (λ) 2 n s=1 rs k=1 Q Xs,k (λ-Xs) k       Xs,- g 0 = - n s=1 X s R Xs,1 + R Xs,2 δ rs≥2 - t ∞,0 ω n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3
(4-30)

The auxiliary matrix Ãα (λ) is given by

Ãα (λ) 1,2 = ω ν (α) ∞,1 λ r∞-4 δ r∞≥4 + r∞-5 j=0   ω ν (α) ∞,r∞-3-j + r∞-4 k=j+1 ν (α) ∞,k-j Q ∞,k   λ j + n s=1 rs j=1   rs k=j ν (α) Xs,k-j Q Xs,k   (λ -X s ) -j
Ãα (λ)

1,1 = 1 2ω L α [ω] -t ∞,1 ν (α) ∞,0 δ r∞,2 + 1 2 -t ∞,0 ν (α) ∞,-1 δ r∞,1 -t ∞,r∞-1 r∞-3 j=0 ν (α) ∞,r∞-2-j λ j -t ∞,r∞-2 r∞-4 j=0 ν (α) ∞,r∞-3-j λ j + r∞-5 j=0 r∞-4-j i=1 ν (α) ∞,i R ∞,j+i λ j + n s=1 rs j=1 rs-j i=0 ν (α) Xs,i R Xs,i+j (λ -X s ) -j Ãα (λ) 2,2 = -Ãα (λ) 1,1 (4-31)
with the extra conditions (4-8) and (4-9) for r ∞ ≤ 2 while coefficients ν

(α) ∞,r∞-2 , ν (α)
∞,r∞-1 are determined by (4-18) for r ∞ ≥ 3 . Finally entry Ãα (λ) Proof. Computations are straightforward for L(λ). The only non-trivial case is r ∞ = 1. For r ∞ = 1, the value of g 0 is given by (A-99) and the expression of L2,1 (λ) follows from Section A.3.1. Computations for [ Ãα (λ)] 1,1 are done in Appendix C.

If the Lax matrices L(λ) and Ãα (λ) have some nice expressions in terms of the geometric Lax coordinates (Q, R), obtaining the Hamiltonian evolutions relatively to these coordinates is not that simple. Indeed, the change of coordinates (Q, P) → (Q, R) is not symplectic so that one cannot use the results on (Q, P) directly. Solving the compatibility equations in these coordinates would require substantial work equivalent to the one done in [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF].

Connection with isospectral deformations

In the previous section, a direct approach to isomonodromic deformations was used to obtain the underlying Hamiltonian structure. The strategy consisted 1. Go to the oper gauge and use appropriate Darboux coordinates to describe the Lax matrices (Proposition (3.1)).

2. Solve the isomonodromic compatibility equation in the oper gauge to obtain the Hamiltonian system (Theorem 3.1).

3. Get back to the initial gauge and perform a time-independent symplectic change of coordinates to write the Hamiltonian and Lax matrices in this gauge (Theorems 4.1 and 4.2).

If this strategy provides a direct way to obtain the symplectic structure, another approach exists in the literature to tackle isomonodromic deformations. Indeed, the historical approach of R. Fuchs (for Fuchsian singularities) pursued by the Japanese school (M. Jimbo, T. Miwa, M. Sato, K. Ueno, etc.) and by the Montréal school (J. Harnad, J. Hurtubise, M. Bertola, etc.) and also by many others consists in using first isospectral deformations and then impose some additional constraints to match them with isomonodromic deformations. It turns out that for Fuchsian singularities the isospectral deformations are identical to the isomonodromic deformations making this strategy a very powerful method to obtain the symplectic structure and the Lax pairs. For non-Fuchsian singularities, the situation is more complicated and the additional constraints are non-trivial. Many cases, including all six Painlevé equations have been dealt with and other case by case studies can be found in the literature. Recently, the connection with isospectral deformations in sl d (C) has been made complete by the Montréal school in [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF]. Their main result (simplified to the sl 2 (C) setting) is summarized in the following theorem.

Theorem 5.1 (Theorems 3.2, 3.3 and 4.6 of [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF]). Let L(λ) ∈ FR,r defining the meromorphic connection ∂ λ Ψ(λ) = L(λ) Ψ(λ). Define (λ + (λ), -λ + (λ)) the eigenvalues of L(λ). The spectral invariants I := (I p,k ) p∈R,k∈ 1,rp-1 are defined using the expansion of the eigenvalues at each pole:

λ + (λ) λ→∞ := r∞-1 j=1 t ∞,j λ j-1 + t ∞,0 λ -1 + r∞-1 j=1 jI ∞,j λ -j-1 + O(λ -r∞-1 ) λ + (λ) λ→Xs := - rs-1 j=1 t Xs,j (λ -X s ) -(j+1) -t Xs,0 (λ -X s ) -1 - rs-1 j=1 jI Xs,j (λ -X s ) j-1 + O((λ -X s ) rs-1 ) , ∀ s ∈ 1, n (5-1)
Let Âα (λ) be the associated rational matrix satisfying L α [ Ψ(λ)] = Âα (λ) Ψ(λ). Let us define the "isopsectral condition" δ

(α) t L(λ) = ∂ λ Âα (λ) (5-2)
where δ (α) t

is the exterior derivative relatively to the irregular times and positions of poles (i.e. we do not derive the Darboux coordinates but only the explicit dependence relatively to the irregular times and positions of poles):

δ (α) t = r∞-1 i=1 α ∞,i δ t ∞,i + n s=1 rs-1 k=1 α Xs,k δ t Xs,k + n s=1 α Xs ∂ Xs (5-3)
If the isospectral condition (5-2) is satisfied then for any (p, k) ∈ R × 1, r p -1 , the spectral invariant I p,k identifies with the Hamiltonian corresponding to the isomonodromic deformation relatively to t p,k . We shall name "isospectral Darboux coordinates", any set of Darboux coordinates (u, v) satisfying the isospectral condition (5-2).

Remark 5.1. In [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF], the authors do not deal with deformations relatively to the position of the finite poles. However, these deformations that correspond to the usual Fuchsian deformations have already been understood for a long time so that only deformations relatively to irregular times remain difficult to describe. For completeness, we shall include the deformations relatively to the position of the poles in this article.

In [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF], the authors observed that the main difficulty of this strategy is that the isospectral condition (5-2) fixes the appropriate Darboux coordinates (up to a trivial time-independent change of coordinates) but that in practice, obtaining some isospectral Darboux coordinates is difficult and left this question opened. We shall address this question providing the link between our geometric Lax coordinates (Q, R) and some isospectral Darboux coordinates (u, v) (Cf. Theorem 5.5).

Expression of det L and verifications on irregular times and monodromies

The formalism presented in [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF] uses the eigenvalues of L(λ) to define the spectral invariants I. In sl 2 (C), one may relate them to det L(λ) and then using the gauge transformation (3-1), relate them to our set H. Indeed, it is straightforward to observe that

det L = det(L + G(∂ λ (G -1 ))) = -L 2,1 -[G(∂ λ (G -1 ))] 2,1 = -L 2,1 + L1,2 ∂ λ L1,1 L1,2 = P2 (λ) - r∞-4 j=0 H ∞,j λ j - n s=1 rs j=1 H Xs,j (λ -X s ) -j + δ r∞≥3 t ∞,r∞-1 λ r∞-3 + g j=1 p j λ -q j + L1,2 ∂ λ L1,1 L1,2 (5-4)
Since L is rational in λ with poles in {∞, X 1 , . . . , X n }, det L is a rational function of λ with poles in {∞, X 1 , . . . , X n }. Thus, we only need to obtain the singular part at each pole in to determine det L. Let us first observe that the term g j=1 p j λ-q j is regular at each pole and hence does not contribute to det L. Thus, we only need to obtain the behavior of L1,2 ∂ λ L1,1 L1,2 at each pole to complete the computation of det L.

Proposition 5.1 (Expression of det L in terms of geometric Darboux coordinates). We have

det L(λ) = P2 (λ) - r∞-4 j=0 H ∞,j λ j - n s=1 rs j=1 H Xs,j (λ -X s ) -j + δ r∞≥3 t ∞,r∞-1 λ r∞-3 + L1,2 ∂ λ L1,1 L1,2 ∞,+ + n s=1 L1,2 ∂ λ L1,1 L1,2 Xs,- (5-5)
with P2 (λ) given by (3-6) and L1,2 (λ), L1,1 (λ) given by either Theorem 4.1 or Theorem 4.3 depending on the choice of Darboux coordinates.

The second purpose of this section is to verify that our irregular times t and monodromies t 0 identify with those of [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF]. We have from Theorem 5.1:

det L(λ) = -λ + (λ) 2 λ→∞ = -   r∞-1 j=0 t ∞,j λ j-1 + r∞-1 j=1 jI ∞,j λ -j-1 + O(λ -r∞-1 )   2 λ→∞ = - r∞-1 i=0 r∞-1 j=0 t ∞,i t ∞,j λ i+j-2 -2 r∞-1 j=1 r∞-1 i=0 jI ∞,j t ∞,i λ i-j-2 + O(λ -2 ) λ→∞ = - 2r∞-4 k=r∞-3 r∞-1 j=k+3-r∞ t ∞,k+2-j t ∞,j λ k - r∞-4 k=0 k+2 j=0 t ∞,k+2-j t ∞,j λ k -2 r∞-4 k=-1 r∞-3-k j=1 jI ∞,j t ∞,j+k+2 λ k + O(λ -2 ) (5-6)
We have from (3-7):

P (2) ∞,k = - r∞-1 j=k+3-r∞ t ∞,j t ∞,k-j+2 , ∀ k ∈ r ∞ -3, 2r ∞ -4
(5-7) so that from Proposition 5.1 we also have:

det L(λ) = P2 (λ) - r∞-4 j=0 H ∞,j λ j - n s=1 rs j=1 H Xs,j (λ -X s ) -j + δ r∞≥3 t ∞,r∞-1 λ r∞-3 + L1,2 ∂ λ L1,1 L1,2 ∞,+ + n s=1 L1,2 ∂ λ L1,1 L1,2 Xs,- λ→∞ = - r∞-1 j=k+3-r∞ t ∞,j t ∞,k-j+2 + δ r∞≥3 t ∞,r∞-1 λ r∞-3 -δ r∞≥3 t ∞,r∞-1 λ r∞-3 +O(λ r∞-4 ) λ→∞ = - r∞-1 j=k+3-r∞ t ∞,j t ∞,k-j+2 + O(λ r∞-4 ) (5-8)
because L1,1 = -t ∞,r∞-1 λ r∞-2 + O(λ r∞-3 ) and L1,2 = ωλ r∞-3 + O(λ r∞-4 ). Thus, for r ∞ ≥ 3, coefficients of order λ r∞-3 up to λ 2r∞-4 match, meaning that the coefficients appearing in the expansion of λ + (λ) at infinity coincide with our irregular times and monodromy at infinity. A similar computation can be carried out for any s ∈ 1, n :

det L(λ) λ→Xs = -   rs-1 j=0 t Xs,j (λ -X s ) -(j+1) + rs-1 j=1 jI Xs,j (λ -X s ) j-1 + O((λ -X s ) rs-1 )   2 - rs-1 i=0 rs-1 j=0 t Xs,j t Xs,i (λ -X s ) -(j+i+2) -2 rs-1 i=0 rs-1 j=1 jI Xs,j t Xs,i (λ -X s ) j-i-2 + O((λ -X s ) -1 ) = - 2rs k=rs+1 rs-1 j=k-rs-1 t Xs,j t Xs,k-j-2 (λ -X s ) -k - rs k=2 k-2 j=0 t Xs,j t Xs,k-j-2 (λ -X s ) -k -2 rs k=2 rs+1-k j=1 jI Xs,j t Xs,k+j-2 (λ -X s ) -k + O((λ -X s ) -1 ) (5-9)
but we have from (3-7):

P (2)
Xs,k = rs-1 j=k-rs-1

t Xs,j t Xs,k-j-2 , ∀k ∈ r s + 1, 2r s (5-10) so that from Proposition (5.1) we also have: up to (λ -X s ) -2rs match, meaning that the coefficients appearing in the expansion of λ + (λ) at λ → X s coincide with our irregular times and monodromy at X s .

det L(λ) = P2 (λ) - r∞-4 j=0 H ∞,j λ j - n s=1 rs j=1 H Xs,j (λ -X s ) -j + δ r∞≥3 t ∞,r∞-1 λ r∞-3 + L1,2 ∂ λ L1,1 L1,2 ∞,+ + n s=1 L1,2 ∂ λ L1,1 L1,2 Xs,- λ→Xs = - 2rs k=rs+1 rs-1 j=k-rs-1 t Xs,j t Xs,k-j-2 (λ -X s ) -k + O((λ -X s ) -rs ) ( 5
Remark 5.2. The situation at infinity for r ∞ ≤ 2 requires special attention. Indeed, we have for r ∞ = 2:

det L(λ) = -t ∞,1 + t ∞,0 λ -1 + O(λ -2 ) 2 λ→∞ = -(t ∞,1 ) 2 -2t ∞,1 t ∞,0 λ -1 + O(λ -2 ) (5-12)
From (5-4) and (3-8) we have:

det L(λ) = -(t ∞,1 ) 2 -   n s=1 H Xs,1 - n j=1 p j   λ -1 + L1,2 ∂ λ L1,1 L1,2 + O(λ -2 ) = -(t ∞,1 ) 2 + (-2t ∞,1 t ∞,0 + t ∞,1 )λ -1 + L1,2 ∂ λ L1,1 L1,2 + O(λ -2 ) = -(t ∞,1 ) 2 -2t ∞,1 t ∞,0 λ -1 + O(λ -2 ) (5-13) because L1,2 ∂ λ L1,1 L1,2 = (ωλ -1 + O(λ -2 ))∂ λ ( - t ∞,r∞-1 ω λ + O(1) = -t ∞,r∞-1 λ -1
. Thus, comparing (5-12) and (5-13) implies that the time t ∞,1 and the monodromy t ∞,0 coincide in both formalisms.

For r ∞ = 1, we have:

det L(λ) = -t ∞,0 λ -1 + O(λ -2 ) 2 = -(t ∞,0 ) 2 λ -2 + O(λ -3 ) (5-14)
From (5-4) and (3-9) we have:

det L(λ) = -   - n s=1 H Xs,1 + n j=1 p j   λ -1 -   n s=1 (H Xs,2 δ rs≥2 + X s H Xs,1 ) - n j=1 q j p j - n s=1 (t Xs,0 ) 2 δ rs=1   λ -2 + L1,2 ∂ λ L1,1 L1,2 + O(λ -3 ) = -(t ∞,0 (t ∞,0 -1)) λ -2 + L1,2 ∂ λ L1,1 L1,2 + O(λ -3 ) (5-15) Since L1,2 ∂ λ L1,1 L1,2 = (ωλ -2 + O(λ -3 ))∂ λ - t ∞,0 ω λ + O(1) = -t ∞,0 λ -2 + O(λ -3 ) (5-16) we end up with det L(λ) = -(t ∞,0 ) 2 λ -2 + O(λ -3 ) (5-17)
Thus, comparing [START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF][START_REF] Chernyakov | Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems[END_REF][START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Fuchs | Sur quelques équations différentielles linéaires du second ordre[END_REF] and [START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF][START_REF] Chernyakov | Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems[END_REF][START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Fuchs | Sur quelques équations différentielles linéaires du second ordre[END_REF][START_REF] Gaiur | Isomonodromic deformations: Confluence, reduction & quantisation[END_REF][START_REF] Gambier | Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes[END_REF][START_REF] Garnier | Solution du problème de Riemann pour les systèmes différentiels linéaires du second ordre[END_REF] implies that the monodromy t ∞,0 coincides in both formalisms.

Hence, we conclude that the irregular times t and monodromies t 0 defined in this article identify exactly with those of [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF]. The next step is to relate the spectral invariants I of [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF] with our set H.

Relation between spectral invariants I and H

The purpose of this section is to relate the spectral invariants I defined by (5-1) with our set H defined by [START_REF] Adams | Dual moment maps into loop algebras[END_REF][START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF]. In order to obtain the relation, let us look at the next orders of the expansion of det L(λ) at each pole. For any s ∈ 1, n , we have from Proposition 5.1:

det L(λ) λ→Xs = - 2rs k=rs+1 rs-1 j=k-rs-1 t Xs,j t Xs,k-j-2 (λ -X s ) -k - rs k=1 H Xs,k (λ -X s ) -k + L1,2 ∂ λ L1,1 L1,2
Xs,- t Xs,j t Xs,k-j-2 (λ -X s ) -k -

+ O(1) (5-18
rs k=2 k-2 j=0 t Xs,j t Xs,k-j-2 (λ -X s ) -k -2 rs k=2 rs+1-k j=1 jI Xs,j t Xs,k+j-2 (λ -X s ) -k + O((λ -X s ) -1 ) (5-19)
Hence, identifying (5-18) and (5-19), we get for all k ∈ 2, r s :

H Xs,k = Res λ→Xs (λ -X s ) k-1 L1,2 (λ)∂ λ L1,1 (λ) L1,2 (λ) + k-2 j=0
t Xs,j t Xs,k-j-2 + 2 rs+1-k j=1 jI Xs,j t Xs,k+j-2

(5-20) These relations may be rewritten into a matrix form:

2M Xs (t)      I Xs,1 2I Xs,2 . . . (r s -1)I Xs,rs-1      =      H Xs,rs H Xs,rs-1 . . . H Xs,2      -            rs-2 j=0 t Xs,j t Xs,rs-j-2 + Res λ→Xs (λ -X s ) rs-1 L1,2 (λ)∂ λ L1,1(λ) L1,2(λ) rs-3 j=0 t Xs,j t Xs,rs-j-3 + Res λ→Xs (λ -X s ) rs-2 L1,2 (λ)∂ λ L1,1(λ) L1,2 (λ)
. . .

(t Xs,0 ) 2 + Res λ→Xs (λ -X s ) L1,2 (λ)∂ λ L1,1(λ) L1,2(λ)            (5-21)
At infinity, for r ∞ ≥ 3, we have from Proposition 5.1:

det L(λ) λ→∞ = - 2r∞-4 k=r∞-3 r∞-1 j=k+3-r∞ t ∞,j t ∞,k-j+2 λ k - r∞-4 k=0 H ∞,k λ k + t ∞,r∞-1 λ r∞-3 + L1,2 ∂ λ L1,1 L1,2 ∞,+ + O(λ -1 ) (5-22)
From (5-6), we have:

det L(λ) λ→∞ = - 2r∞-4 k=r∞-3 r∞-1 j=k+3-r∞ t ∞,k+2-j t ∞,j λ k - r∞-4 k=0 k+2 j=0 t ∞,k+2-j t ∞,j λ k -2 r∞-4 k=-1 r∞-3-k j=1 jI ∞,j t ∞,j+k+2 λ k + O(λ -2 ) (5-23)
For all k ∈ 0, r ∞ -4 , identifying [START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF][START_REF] Chernyakov | Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems[END_REF][START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Fuchs | Sur quelques équations différentielles linéaires du second ordre[END_REF][START_REF] Gaiur | Isomonodromic deformations: Confluence, reduction & quantisation[END_REF][START_REF] Gambier | Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes[END_REF][START_REF] Garnier | Solution du problème de Riemann pour les systèmes différentiels linéaires du second ordre[END_REF][START_REF] Harnad | Dual isomonodromic deformations and moment maps to loop algebras[END_REF][START_REF] Harnad | Hamiltonian Dynamics, Classical R-matrices and Isomonodromic Deformations[END_REF][START_REF] Hurtubise | On the geometry of isomonodromic deformations[END_REF][START_REF] Jimbo | Monodromy preserving deformation of linear ordinary differential equations with rational coefficients[END_REF][START_REF] Jimbo | Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and τ -function[END_REF] and (5-23) implies that:

H ∞,k = -Res λ→∞ λ -k-1 L1,2 (λ)∂ λ L1,1 (λ) L1,2 (λ) + k+2 j=0 t ∞,k+2-j t ∞,j + 2 r∞-3-k j=1 jI ∞,j t ∞,j+k+2 (5-24)
These relations may be rewritten into a matrix form:

2M ∞ (t)      I ∞,1 2I ∞,2 . . . (r ∞ -3)I ∞,r∞-3      =      H ∞,r∞-4 H ∞,r∞-5 . . . H ∞,0      -             r∞-2 j=0 t ∞,r∞-2-j t ∞,j -Res λ→∞ λ -r∞+3 L1,2 (λ)∂ λ L1,1(λ) L1,2(λ) r∞-3 j=0 t ∞,r∞-3-j t ∞,j -Res λ→∞ λ -r∞+4 L1,2 (λ)∂ λ L1,1(λ) L1,2 (λ)
. . .

2 j=0 t ∞,2-j t ∞,j -Res λ→∞ λ -1 L1,2 (λ)∂ λ L1,1(λ) L1,2(λ)            
(5-25)

We may combine (5-21) and (5-25) with Theorem 3.1 to obtain the following statement.

Theorem 5.2 (Relation between Hamiltonians and spectral invariants). The Hamiltonians relatively to the geometric Darboux coordinates (Q, P) are related to the spectral invariants by

      Ham (αt ∞,1 ) (Q, P, t, t 0 ) Ham (αt ∞,2 ) (Q, P, t, t 0 ) . . . (r ∞ -3)Ham (αt ∞,r∞-3 ) (Q, P, t, t 0 )       = 2      I ∞,1 2I ∞,2 . . . (r ∞ -3)I ∞,r∞-3      + (M ∞ (t)) -1             r∞-2 j=0 t ∞,r∞-2-j t ∞,j -Res λ→∞ λ -r∞+3 L1,2 (λ)∂ λ L1,1 (λ) L1,2 (λ) r∞-3 j=0 t ∞,r∞-3-j t ∞,j -Res λ→∞ λ -r∞+4 L1,2 (λ)∂ λ L1,1 (λ) L1,2 (λ)
. . .

2 j=0 t ∞,2-j t ∞,j -Res λ→∞ λ -1 L1,2 (λ)∂ λ L1,1 (λ) L1,2 (λ)             (5-26) and      
Ham (αt Xs,1 ) (Q, P, t, t 0 ) Ham (αt Xs,2 ) (Q, P, t, t 0 ) . . .

(r s -1)Ham (αt Xs,rs-1 ) (Q, P, t, t 0 )       = 2      I Xs,1 2I Xs,2 . . . (r s -1)I Xs,rs-1      + (M Xs (t)) -1            rs-2 j=0 t Xs,j t Xs,rs-j-2 + Res λ→Xs (λ -X s ) rs-1 L1,2 (λ)∂ λ L1,1 (λ) L1,2 (λ) rs-3 j=0 t Xs,j t Xs,rs-j-3 + Res λ→Xs (λ -X s ) rs-2 L1,2 (λ)∂ λ L1,1 (λ) L1,2 (λ)
. . .

(t Xs,0 ) 2 + Res λ→Xs (λ -X s ) L1,2 (λ)∂ λ L1,1 (λ) L1,2 (λ)            (5-27)
for all s ∈ 1, n .

Theorem 5.2 shows that the spectral invariants are indeed related to the Hamiltonian of the system. However, there is no equality between both sets because of the non-trivial terms in the right-hand-sides. These terms are directly related to the gauge transformation G(λ) going from the initial gauge L to the oper gauge L via the additional (∂ λ G)G -1 term that arises in the process. Note that this extra term does not appear when considering isospectral transformations, so it is not a surprise that this term is at the core of the difference between the isospectral world and the isomonodromic world. The main issue is thus to find a way to cancel this extra-term and it is precisely the purpose of [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF] to propose a solution by imposing the isospectral condition. As proved in [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF] and recalled in Theorem 5.1, a possible solution is to impose some additional conditions (eq. (5-2)) on the Lax system. In our setting, this additional condition is equivalent to perform a non-trivial (in the sense time-dependent and not symplectic) change of coordinates that modifies the Hamiltonians to cancel this extra-term.

Isospectral Darboux coordinates

In this section, we look for an explicit change of coordinates (Q, P) → (u, v) so that the coordinates (u, v) satisfy the isospectral condition (5-2). Consequently, for these isospectral coordinates, the spectral invariants I identity with the Hamiltonian of the system. Note that the choice of isospectral coordinates is not unique since any time-independent symplectic change of coordinates from any isospectral coordinates would still provide coordinates satisfying (5-2). Thus, only the explicit dependence in the irregular times is determined by condition (5-2), leaving any symplectic choice of coordinates to determine the time-independent part of the underlying symplectic form. As explained above, the change of coordinates (Q, P) → (u, v) cannot be timeindependent and symplectic since it is required to cancel the extra terms in Theorem 5.2. More specifically, we look for an explicit time dependence so that the new Hamiltonians shall equal the old ones plus the extra term on the r.h.s. of Theorem 5.2. When performing a time-dependent change of coordinates, it is easy to compute the additional terms obtained in the Hamiltonian. On the contrary, our problem goes in the opposite direction: we know the additional terms in the Hamiltonian system that we need to cancel and we want to obtain the time-dependence of the change of coordinates that would correspond to cancel them. To our knowledge, there does not exist any general method to ensure that such a change of coordinates exists nor a constructive method to obtain it in general.

Fortunately, Theorem 4.1 (resp. Theorem 4.3) provides the explicit expressions of the Lax matrices in terms of the coordinates (Q, P) (resp. (Q, R)). Consequently a simple strategy consists in solving directly the isospectral condition (5-2) using these Lax matrices and obtain the required time-dependence for the appropriate change of coordinates.

Differential system for the Q coordinates and definition of the u isospectral coordinates

Let us start with entry (1, 2) of condition (5-2). From Theorem 4.1, we have:

n s=1 rs+1 k=2 (k -1) Q Xs,k-1 δ (α) t [X s ] (λ -X s ) k + n s=1 rs k=1 δ (α) t [Q Xs,k ] (λ -X s ) k + r∞-4 k=0 δ (α) t [Q ∞,k ]λ k + δ (α) t [ω]δ r∞≥3 λ r∞-3 = (r ∞ -4)ω ν (α) ∞,1 λ r∞-5 δ r∞≥4 + r∞-5 j=1 j   ω ν (α) ∞,r∞-3-j + r∞-3 k=j+1 ν (α) ∞,k-j Q ∞,k   λ j-1 - n s=1 rs+1 k=2 (k -1) rs i=k-1 ν (α) Xs,i+1-k Q Xs,i (λ -X s ) -k + reg(λ) (5-28)
where the regular term is O(λ -1 ) when λ → ∞ and O(1) when λ → X s for any s ∈ 1, n . Identifying the coefficients implies the following proposition:

Proposition 5.2. Entry (1, 2) of the condition δ (α) t [ L(λ)] = ∂ λ Ãα (λ) is equivalent to δ (α) t [ω] = -ω ν (α) ∞,-1 δ r∞=1 δ (α) t [X s ] = α Xs,0 , ∀ s ∈ 1, n δ (α) t [Q Xs,1 ] = 0 , ∀ s ∈ 1, n δ (α) t [Q Xs,k ] = -(k -1) rs i=k ν (α) Xs,i+1-k Q Xs,i , ∀(s, k) ∈ 1, n × 2, r s δ (α) t [Q ∞,r∞-4 ] = 0 δ (α) t [Q ∞,r∞-5 ] = (r ∞ -4)ω ν (α) ∞,1 δ (α) t [Q ∞,m ] = (m + 1) ω ν (α) ∞,r∞-4-m + r∞-4 k=m+2 ν (α) ∞,k-m-1 Q ∞,k , ∀ m ∈ 0, r ∞ -6
(5-29)

Proof. Looking at order (λ -X s ) -rs-1 provides:

δ (α) t [X s ] = -ν (α) Xs,0 = α Xs,0 ∀ s ∈ 1, n (5-30) Order (λ -X s ) -k with k ∈ 2, r s provides: (k -1)Q Xs,k-1 δ (α) t [X s ] + δ (α) t [Q Xs,k ] = -(k -1) ν (α) Xs,0 Q Xs,k-1 + rs i=k ν (α) Xs,i+1-k Q Xs,i (5-31)
i.e. using (5-30)

δ (α) t [Q Xs,k ] = -(k -1) rs i=k ν (α) Xs,i+1-k Q Xs,i , ∀(s, k) ∈ 1, n × 2, r s (5-32)
Order (λ -X s ) -1 simply provides δ t [Q Xs,1 ] = 0 for all s ∈ 1, n . Let us look at the situation at infinity depending on the value of r ∞ .

• If r ∞ ≥ 3, order λ r∞-3 provides δ (α)

t [ω] = 0. Then if r ∞ ≥ 4, order λ r∞-4 provides δ (α) t [Q ∞,r∞-4 ] = 0. For r ∞ ≥ 5, order λ r∞-5 provides δ (α) t [Q ∞,r∞-5 ] = (r ∞ -4)ν (α) ∞,1 and for all m ∈ 0, r ∞ -6 order λ m provides δ (α) t [Q ∞,m ] = (m + 1) ν (α) ∞,r∞-4-m + r∞-4 k=m+2 ν (α) ∞,k-m-1 Q ∞,k
(5-33)

• For r ∞ = 2, order λ -1 of (5-28) is equivalent to

n s=1 δ (α) t [Q Xs,1 ] = 0 (5-34) But from (4-2), this is equivalent to δ (α) t [ω] = 0.
• For r ∞ = 1, order λ -1 of (5-28) is equivalent to

n s=1 δ (α) t [Q Xs,1 ] = 0 which is consistent with (4-3). Order λ -2 of (5-28) is equivalent to n s=1 Q Xs,1 δ (α) t [X s ] + n s=1 rs k=1 X s δ (α) t [Q Xs,1 ] + δ (α) t [Q Xs,2 ] = - n s=1 rs i=1 ν (α) Xs,i-1 Q Xs,i (5-35) From (4-3), this is equivalent to δ (α) t [ω] = - n s=1 rs i=1 ν (α) Xs,i-1 Q Xs,i (4-9) = -ω ν (α) ∞,-1
(5-36) A solution to Proposition 5.2 is given by the following theorem.

Theorem 5.3. The conditions δ (α) t [ L1,2 (λ)] = ∂ λ Ã1,2 (λ) are equivalent, for all s ∈ 1, n , to Q Xs,1 = u Xs,1 independent of the irregular times and to solve the (r s -1) × (r s -1) differential system

     tX s,rs -1 0 . . . 0 tX s,rs -2 tX s ,rs-1 0 . . . . . . . . . . . . tX s,1 . . . tX s,rs -2 tX s ,rs-1             1 rs-1 0 . . . 0 0 . . . 0 . . . . . . 0 0 . . . 0 1 1           δt Xs,rs-1 [QX s,rs ] . . . δt Xs,1 [QX s ,rs ] . . . . . . . . . δt Xs,rs -1 [QX s,2 ] . . . δt Xs ,1 [QX s ,2]    =              QX s,
                    1 rs-1 0 . . . 0 0 . . . 0 . . . . . . 0 0 . . . 0 1 1       
(5-37)

whose solutions are of the form:

        QX s ,rs QX s,rs-1 . . . . . . QX s ,2         =          f (Xs)
                 uX s,rs uX s ,rs-1 . . . . . . uX s,2         (5-38)
with all (u Xs,k ) 1≤k≤rs independent of the irregular times. Moreover, we have:

f (Xs) j,j (t Xs,rs-1 ) = (t Xs,rs-1 ) rs-j rs-1 , ∀ j ∈ 1, r s -1 f (Xs) j+1,j (t Xs,rs-2 , t Xs,rs-1 ) = r s -j -1 r s -2 (t Xs,rs-1 ) 1-j rs-1 t Xs,rs-2 , ∀ j ∈ 1, r s -2 f (Xs)
j,1 (t Xs,rs-j , . . . , t Xs,rs-1 ) = t Xs,rs-j , ∀ j ∈ 1, r s -1 (5-39)

At infinity we have for r ∞ ≥ 2 that ω is independent of the irregular times and for r ∞ ≥ 4, Q ∞,r∞-4 = ω u ∞,r∞-4 independent of the irregular times. Moreover, for r ∞ ≥ 5 we have to solve the (r whose solutions are of the form

∞ -4) × (r ∞ -4) differential system         1 0 . . . 0 0 1 0 t∞,r ∞ -3 0 . . . . . . . . . . . . . . . . . . . . . . . . t∞,4 . . . t∞,r ∞ -3 0 1              1 r∞-4 0 . . . 0 0 1 r∞-5 0 . . . . . . . . . . . . 0 . . . . . . 1 1         δt ∞,r∞-3 [Q∞,r ∞ -5] . . . δt ∞,2 [Q∞,r ∞-5 ] . . . δt ∞,r∞-3 [Q∞,0] . . . δt ∞,2 [Q∞,0]    =      ω 0 . . . 0 Q∞,
          ω Q∞,r ∞ -4 Q∞,r ∞ -5 . . . . . . Q∞,0           = ω             1 0 0 . . . . . . 0 0 1 0 . . . . . . f (∞) 3,1 (t∞,r ∞ -3) 0 1 0 0 f (∞) 4,1 (t∞,r ∞-4 , t∞,r ∞ -3) f (∞) 4,2 (t∞,r ∞-3 ) 0 1 . . . . . . . . . . . . . . . 0 f (∞) r∞-2,1 (, t∞,2, . . . , t∞,r ∞-3 ) . . . f (∞) r∞-2,r∞-4 (t∞,r ∞-3 ) 0 1                       1 u∞,r ∞-4 u∞,r ∞-5 . . . . . . u∞,0           (5-41)
with all (u ∞,k ) 0≤k≤r∞-4 independent of the irregular times. Moreover, we have

f (∞) k+2,k (t ∞,r∞-1 ) = r ∞ -3 -k r ∞ -3 t ∞,r∞-3 , ∀ k ∈ 1, r ∞ -4 f (∞) k+3,k (t ∞,r∞-2 , t ∞,r∞-1 ) = r ∞ -4 -k r ∞ -4 t ∞,r∞-4 , ∀ k ∈ 1, r ∞ -5 (5-42)
Finally, for r ∞ = 2, we have the extra condition:

ω = n s=1
u Xs,1 while for r ∞ = 1, we have the extra conditions:

n s=1 u Xs,1 = 0 , ω = n s=1 X s u Xs,1 + n s=1 rs-1 j=1 f (Xs)
rs-1,j (t Xs,j , . . . , t Xs,rs-1 )u Xs,rs+1-j (5-44)

Proof. The proof is done in Appendix D.

Remark 5.3. For r ∞ ≥ 2, the isospectral condition implies that δ (α)

t

[ω] = 0 so that ω can be set to a given non-zero constant (i.e. independent of λ, the irregular times and the positions of the finite poles). Up to a trivial rescaling, this is equivalent to set ω = 1 as done in [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF] or in most of the literature. On the contrary for r ∞ = 1, one observes from Proposition 5.2 and its solution (5-44) that the normalization coefficient ω must depend on the irregular times at infinity in a non-trivial way to ensure the isospectral condition. This difference originates from the fact that for r ∞ = 1, the normalization of L1,2 (λ) is done on a regular order at λ → ∞ (i.e. λ -2 ) that is therefore connected to all the integrable structure at all poles. Hence, imposing the isospectral condition requires that this normalization adapts with the whole integrable structure.

Remark 5.4. We could not obtain a closed form for the expression of the coefficients f

(∞) i,j 1≤i,j≤r∞-2 nor f (Xs) i,j 1≤i,j≤rs-1
. However, we stress that these differential systems are universal and that the corresponding coefficients f

(∞) i,j 1≤i,j≤r∞-2 and f (Xs) i,j 1≤i,j≤rs-1
only depend on the irregular times in a way that may have some interests outside of the scope of the present problem. The exact formulas proposed in Theorem 5.3 provide the explicit expressions up to r s = 4 and r ∞ = 6. Solving each differential system is not obvious and coefficients rapidly get more involved because each line is determined recursively by the formers. The main difficulty is that the r.h.s. of the differential system (5-37) is (after conjugating by the constant matrix) a Toeplitz lower-triangular matrix, while the l.h.s. of the differential system is not a priori. In the end, solving the differential system amounts to choosing a dependence in the irregular times so that the l.h.s. becomes specifically Toeplitz lower-triangular matrix (up to the conjugation by the constant matrix). It turns out that imposing the Toeplitz like condition is non trivial for a linear differential system.

Remark 5.5. The system looks different at infinity because of our choice of normalization at this point. Indeed, imposing the normalization ( L1,2 (λ) = ω λ r∞-3 + O(λ r∞-4 )) is equivalent to freeze a possible coordinate Q ∞,r∞-3 to ω. Moreover, the fact that t ∞,r∞-1 = 1 and t ∞,r∞-2 = 0 is also easily seen in the Toeplitz matrix. Consequently, one could make other choices and immediately adapt the results of Theorem 5.3.

Apart from double poles, one can see that the explicit dependence of the Darboux coordinates Q(t, t 0 ) that is required to satisfy the isospectral condition (5-2) gets substantially involved since solving the differential systems becomes complicated (but can be easily done with an appropriate software). This is not a surprise since as soon as one gets the correct explicit dependence in the irregular times, results of [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF] prove that the Hamiltonians Ham(u, v, t, t 0 ) identify with the spectral invariants I that can be computed from Theorem 5.1 and the explicit expression of L(λ) given by Theorem 4.1. Thus, it seems legitimate that getting the explicit dependence in the irregular times to satisfy the isospetral condition should be as hard as solving the compatibility equations of the Lax system.

Differential system for the R coordinates and definition of the v isospectral coordinates

Adapting the computations done for Theorem 5.3 and Appendix D, we obtain the differential system satisfied by the coordinates R that have been defined in Definition 4.3.

Theorem 5.4. The isospectral condition δ

(α) t L1,1 (λ) = ∂ λ [ Ãα (λ)]
1,1 is equivalent to the fact that for any s ∈ 1, n , the coordinates (R Xs,k ) 1≤k≤rs satisfy the same differential system as (5-37). Thus, for any s ∈ 1, n we have R Xs,1 = v Xs,1 and

        RX s,rs RX s,rs-1 . . . . . . RX s ,2         =          f (Xs)
                 vX s ,rs vX s,rs -1 . . . . . . vX s,2         (5-45)
with all (v Xs,k ) 1≤k≤rs independent of the irregular times. Moreover, for r ∞ = 2 we have the additional relation

n s=1 v Xs,1 = -t ∞,0 (5-46)
and for r ∞ = 1 we have the additional relations

n s=1 v Xs,1 = -t ∞,0 n s=1 X s v Xs,1 + n s=1 rs-1 j=1 f (Xs)
rs-1,j (t Xs,j , . . . , t Xs,rs-1 )v Xs,rs+1-j = -g 0 -

t ∞,0 ω n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3
(5-47)

Similarly, for r ∞ ≥ 4, the coordinates (R ∞,k ) 0≤k≤r∞-4 satisfy the (r

∞ -3) × (r ∞ -3) differential system           1 0 . . . . . . 0 0 1 . . . t∞,r ∞ -3 0 1 . . . . . . . . . . . . . . . 0 t∞,3 . . . t∞,r ∞ -3 0 1                  1 r∞-3 0 . . . 0 0 . . . 0 . . . . . . 0 0 . . . 0 1 1           δt ∞,r∞ -3 [R∞,r ∞ -4] . . . δt ∞,1 [R∞,r ∞ -4] . . . . . . . . . δt ∞,r∞-3 [R∞,0] . . . δt ∞,1 [R∞,0]    =                  -1 0 . . . . . . . . . . . . 0 0 -1 0 . . . R∞,
. . . -1 0 R∞,r ∞,2 R∞,r ∞,3 . . . . . . R∞,r ∞-4 0 -1                         1 r∞-3 0 . . . 0 0 . . . 0 . . . . . . 0 0 . . . 0 1 1       
(5-48)

whose solutions are

        R∞,r ∞-4 R∞,r ∞-5 . . . . . . R∞,0         = -      t∞,r ∞ -3 t∞,r ∞ -4 . . . t∞,1      +             1 0 0 . . . . . . 0 0 1 0 . . . . . . g (∞) 3,1 (t∞,r ∞ -3) 0 1 0 0 g (∞) 4,1 (t∞,r ∞ -4, t∞,r ∞-3 ) g (∞) 4,2 (t∞,r ∞-3 ) 0 1 . . . . . . . . . . . . . . . 0 g (∞) r∞-3,1 (t∞,3, . . . , t∞,r ∞-3 ) . . . g (∞) r∞-3,r∞-4 (t∞,r ∞ -3) 0 1                     v∞,r ∞-4 v∞,r ∞ -5 . . . . . . v∞,0         (5-49)
with all (v ∞,k ) 0≤k≤r∞-4 independent of the irregular times. Moreover, we have:

g (∞) j+2,j (t ∞,r∞-3 ) = r ∞ -4 -j r ∞ -3 t ∞,r∞-3 , ∀ j ∈ 1, r ∞ -5 g (∞) j+3,j (t ∞,r∞-4 , t ∞,r∞-3 ) = r ∞ -5 -j r ∞ -4 t ∞,r∞-4 , ∀ j ∈ 1, r ∞ -6 (5-50)
Proof. The proof is done in Appendix E.

Finally, one may obtain the explicit dependence of the coordinates P in terms of the irregular times so that the isospectral condition (5-2) is realized.

Theorem 5.5. The isospectral condition (5-2) is realized if and only if the coordinates Q satisfy Theorem 5.3 and if the coordinates P are given by (4-27) and (4-28) where the coordinates R satisfy Theorem 5.4.

Proof. It is obvious that the isospectral condition (5-2) implies the specific time dependence of the coordinates (Q, R) providing a necessary condition. Moreover, [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF] proves that a set of coordinates satisfying the isospectral condition always exists providing the sufficient part of the theorem. Another way to obtain the sufficient part of the theorem is to study entry (2, 1) of the isospectral condition. However since this entry is directly expressed from entries (1, 1) and (1, 2) and is computed specifically to match the isomonodromic compatibility equations, it shall not generate new relations on coordinates (Q, R) but rather only corresponds to consistency relations.

Conclusion and outlooks

In this paper we provided several sets of Darboux coordinates to describe the Lax pairs associated to isomonodromic deformations of non-twisted meromorphic connections in sl 2 (C). First, we recalled in Section 3 the results of [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF] and the description of the Lax pair (L, A α ) in the oper gauge using Darboux coordinates (q, p) in relation with the apparent singularities of the quantum curve. This gauge and these coordinates are universal and provide a natural interpretation of the Hamiltonians in terms of interacting particles. However, from the geometric point of view, the fact that the Lax pair and Hamiltonian have singularities at q i ̸ = q j when i ̸ = j is not practical.

We then introduced geometric Darboux coordinates (Q, P) that are related to (q, p) through a time-independent and symplectic change of coordinates. Using these geometric Darboux coordinates, we provided the explicit expressions of the geometric Lax matrices ( L, Ãα ) in Theorem 4.1 and of the associated Hamiltonians in Theorem 4.2. The main interest of these coordinates is to provide expressions directly in the geometric gauge rather than the oper gauge so that the connections with the geometry and previous results in the literature (for example Painlevé cases) are simpler. Moreover, the fact that they are related to (q, p) using a time-independent and symplectic change of coordinates makes the connection with the oper gauge and the corresponding symplectic structure immediate. On the negative side, this makes the formula for L1,1 rather complicated.

In Section 4.4, we defined some geometric Lax coordinates (Q, R) for which the expression of the geometric Lax matrices ( L, Ãα ) is very simple (Theorem 4.3). On the contrary, since the relation with geometric Darboux coordinates (Q, P) is not symplectic, the corresponding Hamiltonians are more difficult to express.

Finally, we defined from the geometric Lax coordinates (Q, R) (Theorems 5.3 and 5.4) some isospectral Darboux coordinates (u, v) that are solutions of the isospectral condition (5-2) of [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF]. In particular, the Hamiltonians corresponding to these coordinates are simply given by the expansion of det L(λ) at each pole and thus are straightforward to obtain. These isospectral coordinates also provide the connection with isospectral deformations that have been extensively studied in the literature. Our main observation in that matter is that for Fuchsian singularities the four sets of coordinates are related in a trivial way. For irregular singularities of low orders (r s ≤ 2 or r ∞ ≤ 6), isospectral Darboux coordinates identify with geometric Darboux coordinates (Q, P) when choosing the appropriate normalization of the Lax pairs. Nevertheless, this is no longer the case for irregular singularities of higher orders where the relations between both sets become non-trivial. This observation may explain why the isospectral deformations strategy has been successful in some numerous case-by-case analysis but had not been generalized so far. In sl 2 (C), our results provide an answer to the question left opened in [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF] to find isospectral coordinates.

There are several extensions of our work that would deserve investigations

• The first natural extension consists in obtaining similar results in sl d (C) with d ≥ 3. Indeed, the isospectral strategy developed in [START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF] is valid for any d ≥ 2. It seems to the authors that using properties of the gauge transformation from the geometric gauge to the oper gauge in a smart way should be sufficient to obtain the expression of the Lax pairs and of the Hamiltonian system and we let this question for future works.

• The differential systems solving the isospectral condition (5-2) (Theorems 5.3 and 5.4) amount to transform a matrix of derivative into a lower-triangular Toeplitz invertible matrix. It is worth noticing that m×m lower-triangular Toeplitz matrices are a commuting subalgebra of dimension m of GL m (C) (that can be extended to a subfield by considering lower-triangular Toeplitz matrices with non-zero diagonal entries). From this point of view, the isospectral condition (5-2) seems equivalent to transform a problem from the algebra of matrices to a commuting subalgebra (of lower-triangular Toeplitz matrices). This observation would definitely deserve additional investigations from a purely algebraic point of view and possible interpretation from a geometric point of view.

• In [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlevé 1 hierarchy[END_REF], the authors derived the expression of the Lax matrices ( L, Ãα ) for the case of twisted meromorphic connections with an irregular pole at infinity with a nilpotent leading order (i.e. an element of F {∞},r∞ but not in F{∞},r∞ ). Hence, one could easily write down the isospectral condition for this case (i.e. only Section 5 of the present work is still missing) and obtain the isospectral coordinates in this twisted case. It would be interesting to see if the differential systems obtained in this twisted case are similar to the ones obtained in this article as well as the potential differences on the isospectral coordinates.

A Proof of Theorem 4.1

A.1 Entry (1, 2)

The choice of coordinates Q immediately provides the expression for L1,2 (λ). Moreover, the gauge transformation (3-1) implies that for r ∞ ≥ 3

G(λ) Ãα (λ) = A α (λ)G(λ) -L α [G(λ)] (A-1) In particular, [ Ãα (λ)] 1,2 = L1,2 (λ)[A α (λ)] 1,2 . (A-2)
Since we know the expansion of [A α (λ)] 1,2 at each pole we get for r ∞ ≥ 3:

[ Ãα (λ)] 1,2 λ→∞ = n s=1 rs k=1 Q Xs,k (λ -X s ) k + r∞-4 k=0 Q ∞,k λ k + ω δ r∞≥3 λ r∞-3 r∞-3 i=1 ν (α) ∞,i λ -i + O λ -r∞+2 = ω δ r∞≥3 r∞-3 i=1 ν (α) ∞,i λ r∞-3-i + r∞-4 k=0 r∞-3 i=1 Q ∞,k ν (α) ∞,i λ k-i + O λ -1 = ω δ r∞≥3 r∞-4 j=0 ν (α) ∞,r∞-3-j λ j + r∞-5 j=0 r∞-4 k=j+1 Q ∞,k ν (α) ∞,k-j λ j + O λ -1 = ω ν (α) ∞,1 δ r∞≥4 λ r∞-4 + r∞-5 j=0   ω ν (α) ∞,r∞-3-j + r∞-4 k=j+1 ν (α) ∞,k-j Q ∞,k   λ j + O(λ -1 ) (A-3)
For r ∞ = 2 we have:

[ Ãα (λ)] 1,2 λ→∞ = ωλ -1 + O(λ -2 ) ν (α) ∞,0 + O(λ -1 ) = ω ν (α) ∞,0 λ -1 + O(λ -2 ) (A-4)
For r ∞ = 1, we have:

[ Ãα (λ)] 1,2 λ→∞ = ωλ -2 + n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 λ -3 + O(λ -4 ) ν (α) ∞,-1 λ + ν (α) ∞,0 + O(λ -1 ) = ω ν (α) ∞,-1 λ -1 + n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 ν (α) ∞,-1 + ω ν (α) ∞,0 λ -2 +O(λ -3 ) (A-5)
In the same way, for all s ∈ 1, n :

[ Ãα (λ)] 1,2 λ→Xs = n s ′ =1 r s ′ k=1 Q X s ′ ,k (λ -X s ′ ) k + r∞-4 k=0 Q ∞,k λ k + ω δ r∞≥3 λ r∞-3 rs-1 i=0 ν (α) Xs,i (λ -X s ) i + O ((λ -X s ) rs )
But the l.h.s. is a rational function of λ with poles in {∞, X 1 , . . . , X n }. It can be rewritten as

-ω g j̸ =i (λ -q j ) n s=1 (λ -X s ) rs = -ω g j=1 (λ -q j ) (λ -q i ) n s=1 (λ -X s ) rs (A-12)
We may expand the last quantity at each pole:

-ω g j=1 (λ -q j ) (λ -q i ) n s=1 (λ -X s ) rs λ→∞ = - n s=1 rs k=1 Q Xs,k (λ -X s ) k + r∞-4 k=0 Q ∞,k λ k + ω δ r∞≥3 λ r∞-3 ∞ r=0 q r i λ -1-r = - r∞-4 k=0 ∞ r=0 Q ∞,k q r i λ k-1-r -ω r∞-4 r=0 q r i λ r∞-4-r + O(λ -1 ) m=k-1-r = - r∞-5 m=0 r∞-4 k=m+1 Q ∞,k q k-1-m i λ m -ω r∞-4 m=0 q r∞-4-m i λ m + O(λ -1 ) (A-13)
Note that the expression is also valid for r ∞ ≤ 3 in which case the r.h.s. is only O(λ -1 ). For any s ∈ 1, n , we also have:

-ω g j=1 (λ -q j ) (λ -q i ) n s=1 (λ -X s ) rs λ→Xs = - n s=1 rs k=1 Q Xs,k (λ -X s ) k + r∞-4 k=0 Q ∞,k λ k + ω δ r∞≥3 λ r∞-3 ∞ r=0 (-1) r (X s -q i ) -(r+1) (λ -X s ) r = - rs k=1 ∞ r=0 Q Xs,k (-1) r (X s -q i ) -(r+1) (λ -X s ) r-k + O(1) m=k-r = - rs m=1 rs k=m Q Xs,k (-1) k-m (X s -q i ) m-k-1 (λ -X s ) -m + O(1) = rs m=1 rs k=m Q Xs,k (q i -X s ) m-k-1 (λ -X s ) -m + O(1) (A-14)
Thus we get:

-ω g j=1 (λ -q j ) (λ -q i ) n s=1 (λ -X s ) rs = -ω r∞-4 m=0 q r∞-4-m i λ m - r∞-5 m=0 r∞-4 j=m+1 Q ∞,j q j-1-m i λ m + n s=1 rs m=1 rs j=m Q Xs,j (q i -X s ) m-j-1 (λ -X s ) -m (A-15)
For r ∞ = 2, the l.h.s. is O(λ -2 ) at infinity and for r ∞ = 1, the l.h.s. is O(λ -3 ) at infinity so that we have the additional relations for all i ∈ 1, g :

0 = n s=1 rs j=1 Q Xs,j (q i -X s ) -j , if r ∞ = 2 0 = n s=1 rs j=1 Q Xs,j (q i -X s ) -j and 0 = n s=1 rs j=2 Q Xs,j (q i -X s ) 1-j + n s=1 rs j=1 X s Q Xs,j (q i -X s ) -j , if r ∞ = 1 (A-16)
Finally, we may identify the coefficients at each pole with (A-11) to determine ∂ q i [Q p,k ] ending the proof of Lemma A.1. Note that the additional relations (A-16) are consistent with the additional constraints (4-2) and (4-3). The next step is to provide an expression for -

Q(λ) n s=1 (λ-Xs) rs . Lemma A.2. We have - Q(λ) n s=1 (λ -X s ) rs = -ω δ r∞≥4 P ∞,r∞-4 - r∞-5 m=0 P ∞,m   λ r∞-4-m + ω r∞-4 j=m+1 Q ∞,j λ j-1-m   + n s=1 rs m=1 P Xs,m   rs j=m (λ -X s ) m-j-1 Q Xs,j   = -ω r∞-4 k=0 P ∞,r∞-4-k λ k - r∞-5 k=0 r∞-5-k m=0 P ∞,m Q ∞,k+1+m λ k + n s=1 rs k=1 rs+1-k m=1 P Xs,m Q Xs,k+m-1 (λ -X s ) -k (A-17)
Proof. Let r ∞ ≥ 3. By definition, both sides are rational function of λ with poles of degree r ∞ -4 at infinity and r s at X s for all s ∈ 1, n . Moreover, for any i ∈ 1, g we have Q(q i ) = -p i n s=1

(q i -X s ) rs so that

- Q(q i ) n s=1 (q i -X s ) rs = p i Def. 4.1 = δ r∞≥4 P ∞,r∞-4 ∂ q i [Q ∞,r∞-4 ] + r∞-5 k=0 P ∞,k ∂ q i [Q ∞,k ] + n s=1 rs k=1 P Xs,k ∂ q i [Q Xs,k ] Lemma A.1 = -ω δ r∞≥4 P ∞,r∞-4 - r∞-5 k=0 P ∞,k   ω q r∞-4-k i + r∞-4 j=k+1 Q ∞,j q j-1-k i   + n s=1 rs k=1 P Xs,k   rs j=k (q i -X s ) k-j-1 Q Xs,j   (A-18)
We recognize the r.h.s. of Lemma A.2 evaluated at λ = q i . Thus, for all i ∈ 1, g , both sides coincide at λ = q i . Multiplying, by n s=1 (λ -X s ) rs on both sides, we get polynomials of order g -1 that coincide at g distinct values so that they are equal ending the proof of Lemma

• For r ∞ = 2, we have from (4-8)

Ã(α) 1,2 (λ) = ω ν (α) ∞,0 λ -1 + O(λ -2 ) so that ∂ λ Ã(α) 1,2 (λ) = -ω ν (α) ∞,0 λ -2 +O(λ -3 ). We have also L1,2 (λ) = ωλ -1 + n s=1 X s Q Xs,1 + Q Xs,2 λ -2 +O(λ -3 ) and [A α ] 1,1 = c ∞,0 + a 1 λ + O(λ -2 ) so that L α [ L1,2 (λ)] = 2ωc ∞,0 λ -1 + -ω ν (α) ∞,0 + 2ωa 1 + 2c ∞,0 n s=1 X s Q Xs,1 + Q Xs,2 λ -2 +O(λ -3 ) (A-35)
Identifying coefficient λ -1 at infinity provides

L α [ω] = 2ωc ∞,0 ⇒ c ∞,0 = 1 2ω L α [ω] (A-36) Identifying coefficient λ -2 at infinity provides n s=1 L α [X s Q Xs,1 + Q Xs,2 ] = 1 ω L α [ω] n s=1 X s Q Xs,1 + Q Xs,2 -ω ν (α) ∞,0 + 2ωa 1 (A-37)
• For r ∞ = 1, we have from (4-9)

Ã(α) 1,2 (λ) = ω ν (α) ∞,-1 λ -1 + ω ν (α) ∞,0 + ν (α) ∞,-1 n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 λ -2 +O(λ -3 ) (A-38) so that ∂ λ Ã(α) 1,2 (λ) = -ω ν (α) ∞,-1 λ -2 -2 ω ν (α) ∞,0 + ν (α) ∞,-1 n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 λ -3 + O(λ -4 ) (A-39) Moreover, L1,2 (λ) = ωλ -2 + n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 λ -3 +O(λ -4 ) and A (α) 1,1 (λ) = c ∞,0 + a 1 λ + O(λ -2 ) so that L α [ L1,2 (λ)] = 2ωc ∞,0 -ω ν (α) ∞,-1 λ -2 +2 ωa 1 -ω ν (α) ∞,0 + (c ∞,0 -ν (α) ∞,-1 ) n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 λ -3 +O(λ -4 ) (A-40)
Identifying coefficient λ -2 at infinity provides

L α [ω] = 2ωc ∞,0 -ω ν (α) ∞,-1 ⇒ c ∞,0 = 1 2ω L α [ω] + 1 2 ν (α) ∞,-1 (A-41) Identifying coefficient λ -3 at infinity provides n s=1 L α [X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 ] = 2ω a 1 -2ω ν (α) ∞,0 + 1 ω L α [ω] -ν (α) ∞,-1 n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 (A-42)
We now use the fact that L2,1 is a rational function of λ with poles only in {∞, X 1 , . . . , X n }. Thus, we may simply study the behavior of each term at these poles. Let us first discuss the term L 2,1 (λ) L1,2 (λ) . We recall that L1,2 (λ) = ω δ r∞≥3 λ r∞-3

+ r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k and that from (3-5) L 2,1 (λ) = -P2 (λ) + r∞-4 j=0 H ∞,j λ j + n s=1 rs j=1 H Xs,j (λ -X s ) -j -λ r∞-3 δ r∞≥3 - g j=1 p j λ -q j (A-53)
Thus we have

L 2,1 (λ) L1,2 (λ) = L 2,1 (λ) ω δ r∞≥3 λ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k λ→Xs = -P2 (λ) + O ((λ -X s ) -rs ) ω δ r∞≥3 λ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k λ→Xs = -P2 (λ) ω δ r∞≥3 λ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k + O(1) λ→Xs =       2rs j=rs+1 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 (λ -X s ) -j ω δ r∞≥3 λ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k       Xs,- + O(1) (A-54)
At infinity, we have

• For r ∞ ≥ 3: L 2,1 (λ) L1,2 (λ) λ→∞ = -P2 (λ) -t ∞,r∞-1 λ r∞-3 + O(λ r∞-4 ) ω λ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k λ→∞ = - t ∞,r∞-1 ω +       2r∞-4 j=r∞-3 2r∞-4-j m=0 t ∞,r∞-1-m t ∞,j+m-r∞+3 λ j ω λ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k       ∞,+ +O(λ -1 ) (A-55)
• For r ∞ = 2: From (3-8) we have:

L 2,1 (λ) λ→∞ = (t ∞,1 ) 2 + (2t ∞,1 t ∞,0 -t ∞,1 )λ -1 + O(λ -2 ) (A-56) while L1,2 (λ) λ→∞ = ωλ -1 + n s=1 (Q Xs,2 + X s Q Xs,1 ) λ -2 + O(λ -3 ) (A-57)
Thus, we obtain

L 2,1 (λ) L1,2 (λ) λ→∞ = (t ∞,1 ) 2 ω λ - (t ∞,1 ) 2 ω 2 n s=1 (Q Xs,2 + X s Q Xs,1 ) + 2t ∞,1 t ∞,0 -t ∞,1 ω + O(λ -1 )
(A-58) Moreover, we have:

- ( L1,1 (λ)) 2 L1,2 (λ) λ→∞ = - t ∞,1 + t ∞,0 λ -1 + O(λ -2 ) 2 ωλ -1 + n s=1 X s Q Xs,1 + Q Xs,2 λ -2 + O(λ -3 ) = - (t ∞,1 ) 2 ω λ - 2t ∞,0 t ∞,1 ω + (t ∞,1 ) 2 ω 2 n s=1 X s Q Xs,1 + Q Xs,2 + O(λ -1 ) (A-59)
• For r ∞ = 1: We have from (3-9)

L 2,1 (λ) λ→∞ = (t ∞,0 (t ∞,0 -1)) λ -2 + O(λ -3 ) (A-60) while L1,2 (λ) λ→∞ = ωλ -2 + O(λ -3 ) (A-61) Thus, we obtain L 2,1 (λ) L1,2 (λ) λ→∞ = t ∞,0 (t ∞,0 -1) ω + O(λ -1 ) (A-62)
Moreover, we have:

- ( L1,1 (λ)) 2 L1,2 (λ) λ→∞ = - t ∞,0 λ -1 + O(λ -2 ) 2 ωλ -2 + O(λ -3 ) = - (t ∞,0 ) 2 ω + O(λ -1 ) (A-63)
Let us now discuss the term -∂ λ L1,1 (λ) L1,2 (λ) . Since both L1,1 (λ) and L1,2 (λ) are singular at λ → X s of order (λ -X s ) rs , it is obvious that

-∂ λ L1,1 (λ) L1,2 (λ) λ→Xs = O(1) , ∀ s ∈ 1, n (A-64)
Because of the normalization at infinity, we have:

-∂ λ L1,1 (λ) L1,2 (λ) λ→∞ = -∂ λ -t ∞,r∞-1 λ r∞-2 + O(λ r∞-3 ) ωλ r∞-3 + O(λ r∞-4 ) λ→∞ = t ∞,r∞-1 ω + O(λ -2 ) (A-65) so that the only contribution of -∂ λ L1,1 (λ) L1,2 (λ) in (A-52) is t ∞,r∞-1 ω
. Regrouping the contributions of each term at each pole, we end up with formulas of Theorem 4.1.

A.3.2 The special case of r ∞ = 1

Finally, let us discuss the special case of r ∞ = 1. The previous contributions indicate that we may rewrite L2,1 (λ) as

L2,1 (λ) = n s=1       2rs j=rs+1 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 (λ -X s ) -j -L1,1 (λ) 2 L1,2 (λ)       Xs,- + n s=1 2(t ∞,0 λ + g 0 ) ω L1,1 (λ) - (t ∞,0 λ + g 0 ) 2 ω 2 L1,2 (λ)
Xs,-

(A-66)
Let us now observe that

2(t ∞,0 λ+g 0 ) ω L1,1 (λ) - (t ∞,0 λ+g 0 ) 2 ω 2
L1,2 (λ) is a rational function of λ with poles in R. At infinity we have

2(t ∞,0 λ + g 0 ) ω L1,1 (λ) - (t ∞,0 λ + g 0 ) 2 ω 2 L1,2 (λ) λ→∞ = - (t ∞,0 ) 2 ω + O(λ -1 ) (A-67) so that we get 2(t ∞,0 λ + g 0 ) ω L1,1 (λ) - (t ∞,0 λ + g 0 ) 2 ω 2 L1,2 (λ) = n s=1 (t ∞,0 λ + g 0 ) ω L1,1 (λ) - (t ∞,0 λ + g 0 ) 2 ω 2 L1,2 (λ) Xs,- - (t ∞,0 ) 2 ω (A-68)
Thus, we end up with

L2,1 (λ) = (t ∞,0 ) 2 ω + n s=1       2rs j=rs+1 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 (λ -X s ) -j -L1,1 (λ) 2 L1,2 (λ)       Xs,- + 2(t ∞,0 λ + g 0 ) ω L1,1 (λ) - (t ∞,0 λ + g 0 ) 2 ω 2 L1,2 (λ) (A-69)
Let us now obtain the expression for g 0 . We first observe that L1,1 (λ) is independent of g 0 by definition. Moreover, the gauge transformation and the normalization at infinity imply that

g 0 = 1 ω 1 2 -t ∞,0 n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 - 1 2t ∞,0 Res λ→∞ λ 2 L 2,1 (λ) (A-70) and L 2,1 (λ) = ( L1,1 ) 2 + L1,2 (λ) L2,1 (λ) + ∂ λ L1,1 (λ) -L1,1 (λ) ∂ λ L1,2 (λ) L1,2 (λ) = ( L1,1 (λ)) 2 + ∂ λ L1,1 (λ) - t ∞,0 ω L1,2 (λ) -L1,1 (λ) ∂ λ L1,2 (λ) L1,2 (λ) + L1,2 (λ) L2,1 (λ) - (t ∞,0 λ + g 0 ) ω L1,2 (λ) L1,1 (λ) + (t ∞,0 λ + g 0 ) 2 ω 2 L1,2 (λ) 2 (A-71)
Using the specific formula for L2,1 (λ) given by (A-69), we end up with

L 2,1 (λ) = ( L1,1 (λ)) 2 + ∂ λ L1,1 (λ) - t ∞,0 ω L1,2 (λ) -L1,1 (λ) ∂ λ L1,2 (λ) L1,2 (λ) + (t ∞,0 ) 2 ω L1,2 (λ) + L1,2 (λ) n s=1       2rs j=rs+1 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 (λ -X s ) -j -L1,1 (λ) 2 L1,2 (λ)       Xs,- (A-72)
so that

g 0 = 1 ω 1 2 -t ∞,0 n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 - 1 2t ∞,0 Res λ→∞ λ 2 ( L1,1 (λ)) 2 + ∂ λ L1,1 (λ) - t ∞,0 ω L1,2 (λ) -L1,1 (λ) ∂ λ L1,2 (λ) L1,2 (λ) + (t ∞,0 ) 2 ω L1,2 (λ) + L1,2 (λ) n s=1       2rs j=rs+1 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 (λ -X s ) -j -L1,1 (λ) 2 L1,2 (λ)       Xs,- (A-73) A.3.3 Computation of [ Ãα (λ)] 2,1
Entries (2, 1) and (2, 2) of the gauge transformation (A-1) imply that

L1,1 (λ)[ Ãα (λ)] 1,1 + L1,2 (λ)[ Ãα (λ)] 2,1 = [A α (λ)] 2,1 + L1,1 (λ)[A α (λ)] 2,2 -L α [ L1,1 (λ)] L1,1 (λ)[ Ãα (λ)] 1,2 -L1,2 (λ)[ Ãα (λ)] 1,1 = [A α (λ)] 2,2 L1,2 (λ) -L α [ L1,2 (λ)] (A-74) Using (3-21) we get [ Ãα (λ)] 2,1 = -L α L1,1 (λ) L1,2 (λ) + ∂ λ [A α (λ)] 1,1 L1,2 (λ) + [A α (λ)] 1,2 L 2,1 (λ) L1,2 (λ) + L1,1 (λ) L1,2 (λ) L1,1 (λ) L1,2 (λ) [ Ãα (λ)] 1,2 -2[ Ãα (λ)] 1,1 (A-75)
Since [ Ãα (λ)] 2,1 is a rational function of λ with poles in {∞, X 1 , . . . , X n }, we only need to evaluate the contributions at each pole.

A.3.4 Term -L α L1,1 (λ) L1,2 (λ)
Let us discuss the term -L α L1,1 (λ) L1,2 (λ) . We recall that L1,1 (λ) L1,2 (λ)

λ→Xs = O(1) ∀ s ∈ 1, n (A-76)
At infinity, we have:

L1,1 (λ) L1,2 (λ) λ→∞ = -t ∞,r∞-1 λ -g 0 ω + O(λ -1 ) (A-77)
In all cases, the contribution of -t ∞,r∞-1 λ to -L α L1,1 (λ) L1,2 (λ) is null because t ∞,r∞-1 is either a monodromy parameter (for r ∞ = 1) or fixed to 1 (for r ∞ ≥ 2) because of Proposition 2.1. Thus, we end up with the fact that -L α L1,1 (λ) L1,2 (λ) only contributes with either

• - t ∞,r∞-1 ω 2 L α [ω]λ -L α Q ∞,r∞-4 ω 2 for r ∞ ≥ 4 • - t ∞,2 ω 2 L α [ω]λ - n s=1 L α Q Xs,1 ω 2 for r ∞ = 3 • - t ∞,1 ω 2 L α [ω]λ - n s=1 L α Q Xs,2 +XsQ Xs,1 ω 2 for r ∞ = 2 • - t ∞,0 ω 2 L α [ω]λ + L α g 0 ω for r ∞ = 1.
Note in particular that in all four cases, the contribution is linear in λ. We may now use (A-31), (A-34), (A-37) and (A-42) to compute the contribution for each case.In order to use these formulas, we need to express a 1 which is the coefficient of order λ

-1 of [A α ] 1,1 (λ) at λ → ∞.
From (A-2) and (A-43), we have

A (α) 1,1 (λ) = Ã(α) 1,1 (λ) - L(α) 1,1 (λ) L(α) 1,2 (λ) Ã(α) 1,2 (λ) (A-78) so that a 1 = -Res λ→∞ Ã(α) 1,1 (λ) - L(α) 1,1 (λ) L(α) 1,2 (λ) Ã(α) 1,2 (λ) (A-79)
Thus we get that the contribution of -L α

L1,1 (λ) L1,2 (λ) is • - t ∞,r∞-1 ω 2 L α [ω]λ + 2 ω Res λ→∞ Ã(α) 1,1 (λ) - L(α) 1,1 (λ) L(α) 1,2 (λ) Ã(α) 1,2 (λ) + 1 ω 3 Q ∞,r∞-4 L α [ω] for r ∞ ≥ 4. • - t ∞,2 ω 2 L α [ω]λ + 2 ω Res λ→∞ Ã(α) 1,1 (λ) - L(α) 1,1 (λ) L(α) 1,2 (λ) Ã(α) 1,2 (λ) + 1 ω 3 n s=1 Q Xs,1 L α [ω] for r ∞ = 3. • - t ∞,1 ω 2 L α [ω]λ + 1 ω ν (α) ∞,0 + 2 ω Res λ→∞ Ã(α) 1,1 (λ) - L(α) 1,1 (λ) L(α) 1,2 (λ) Ã(α) 1,2 (λ) + 1 ω 3 n s=1 X s Q Xs,1 + Q Xs,2 L α [ω] for r ∞ = 2. • - t ∞,0 ω 2 L α [ω]λ + L α g 0 ω for r ∞ = 1. A.3.5 Term ∂ λ [Aα(λ)] 1,1 L1,2 (λ)
Let us now discuss the term

∂ λ [Aα(λ)] 1,1 L1,2 (λ)
in (A-75). This quantity is obviously regular at λ → X s for all s ∈ 1, n . At infinity, we have

[A α (λ)] 1,1 = O(1) so that ∂ λ [A α (λ)] 1,1 = O(λ -2 ). Since L1,2 (λ) = ωλ r∞-3 + O(λ r∞-4
), we only get a contribution for r ∞ = 1 and it is constant in λ.

A.3.6 Term

[A α (λ)] 1,2 L 2,1 (λ) L1,2 (λ) Let us now discuss the term [A α (λ)] 1,2 L 2,1 (λ)
L1,2 (λ) in (A-75). For s ∈ 1, n , the behavior is given by (A-54) and :

[A α (λ)] 1,2 L 2,1 (λ) L1,2 (λ) λ→Xs =       2rs j=rs+1 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 (λ -X s ) -j ω δ r∞≥3 λ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k + O(1)       rs-1 i=0 ν (α) Xs,i (λ -X s ) i + O ((λ -X s ) rs ) λ→Xs = rs-1 i=0 2rs j=rs+1 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 ν (α) Xs,i (λ -X s ) i-j ω δ r∞≥3 λ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k + O(1) k=j-i = 2rs k=rs+1 2rs j=k 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 ν (α) Xs,j-k (λ -X s ) -k ω δ r∞≥3 λ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k + O(1) (A-80)
The situation at infinity depends on the value of r ∞ .

• If r ∞ ≥ 3, we have from (A-55) and (3-23):

[A α (λ)] 1,2 L 2,1 (λ) L1,2 (λ) λ→∞ =      2r∞-4 j=r∞-2 2r∞-4-j m=0 t ∞,r∞-1-m t ∞,j+m-r∞+3 λ j ω λ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k + O(1)      r∞-1 i=1 ν (α) ∞,i λ i + O λ -r∞ λ→∞ = r∞-1 i=1 2r∞-4 j=r∞-2 2r∞-4-j m=0 t ∞,r∞-1-m t ∞,j+m-r∞+3 ν (α) ∞,i λ j-i ω λ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k + O(λ -1 ) k=j-i = 2r∞-5 k=r∞-3 2r∞-4 j=k+1 2r∞-4-j m=0 t ∞,r∞-1-m t ∞,j+m-r∞+3 ν (α) ∞,j-k λ k ω λ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k + O(λ -1 ) (A-81)
• If r ∞ = 2, we have from (A-58) and :

[Aα(λ)]1,2 L2,1(λ) L1,2(λ) λ→∞ = (t∞,1) 2 ω λ - (t∞,1) 2 ω 2 n s=1 (QX s,2 + XsQX s,1 ) + 2t∞,1t∞,0 -t∞,1 ω + O(λ -1 ) ν (α) ∞,0 + ν (α) ∞,1 λ -1 + O(λ -2 )
• For r ∞ ≥ we have:

[ Ãα (λ)] 2,1 = - t ∞,r∞-1 ω 2 L α [ω]λ - 2 ω Res λ→∞ Ã(α) 1,1 (λ) - L(α) 1,1 (λ) L(α) 1,2 (λ) Ã(α) 1,2 (λ) + 1 ω 3 Q ∞,r∞-4 L α [ω] + L1,1 (λ) L1,2 (λ) L1,1 (λ) L1,2 (λ) [ Ãα (λ)] 1,2 -2[ Ãα (λ)] 1,1 ∞,+ + n s=1       2rs k=rs+1 2rs j=k 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 ν (α) Xs,j-k (λ -X s ) -k ωλ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k       Xs,- +       2r∞-5 k=r∞-3 2r∞-4 j=k+1 2r∞-4-j m=0 t ∞,r∞-1-m t ∞,j+m-r∞+3 ν (α) ∞,j-k λ k ωλ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k       ∞,+ + n s=1 L1,1 (λ) L1,2 (λ) L1,1 (λ) L1,2 (λ) [ Ãα (λ)] 1,2 -2[ Ãα (λ)] 1,1 Xs,- (A-91)
• For r ∞ = we have:

[ Ãα (λ)] 2,1 = - t ∞,2 ω 2 L α [ω]λ - 2 ω Res λ→∞ Ã(α) 1,1 (λ) - L(α) 1,1 (λ) L(α) 1,2 (λ) Ã(α) 1,2 (λ) + 1 ω 3 n s=1 Q Xs,1 L α [ω] + L1,1 (λ) L1,2 (λ) L1,1 (λ) L1,2 (λ) [ Ãα (λ)] 1,2 -2[ Ãα (λ)] 1,1 ∞,+ + n s=1       2rs k=rs+1 2rs j=k 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 ν (α) Xs,j-k (λ -X s ) -k ωλ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k       Xs,- +       2r∞-5 k=r∞-3 2r∞-4 j=k+1 2r∞-4-j m=0 t ∞,r∞-1-m t ∞,j+m-r∞+3 ν (α) ∞,j-k λ k ωλ r∞-3 + r∞-4 k=0 Q ∞,k λ k + n s=1 rs k=1 Q Xs,k (λ-Xs) k       ∞,+ + n s=1 L1,1 (λ) L1,2 (λ) L1,1 (λ) L1,2 (λ) [ Ãα (λ)] 1,2 -2[ Ãα (λ)] 1,1 Xs,- (A-92)
• For r ∞ = we have:

[ Ãα (λ)] 2,1 = - t ∞,1 ω 2 L α [ω]λ + 1 ω ν (α) ∞,0 - 2 ω Res λ→∞ Ã(α) 1,1 (λ) - L(α) 1,1 (λ) L(α) 1,2 (λ) Ã(α) 1,2 (λ) + 1 ω 3 n s=1 X s Q Xs,1 + Q Xs,2 L α [ω] + (t ∞,1 ) 2 ω ν (α) ∞,0 λ + (t ∞,1 ) 2 ω ν (α) ∞,1 +ν (α) ∞,0 - (t ∞,1 ) 2 ω 2 n s=1 (Q Xs,2 + X s Q Xs,1 ) + 2t ∞,1 t ∞,0 -t ∞,1 ω + L1,1 (λ) L1,2 (λ) L1,1 (λ) L1,2 (λ) [ Ãα (λ)] 1,2 -2[ Ãα (λ)] 1,1 ∞,+ + n s=1       2rs k=rs+1 2rs j=k 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 ν (α) Xs,j-k (λ -X s ) -k n s=1 rs k=1 Q Xs,k (λ-Xs) k       Xs,- + n s=1 L1,1 (λ) L1,2 (λ) L1,1 (λ) L1,2 (λ) [ Ãα (λ)] 1,2 -2[ Ãα (λ)] 1,1 Xs,- (A-93)
Using previous results, we may evaluate the contribution at infinity and observe that it is null.

• For r ∞ = 1, we get from Section A.3.4, (A-86) and (A-90) that [ Ãα (λ)] 2,1 behaves at infinity like

[ Ãα (λ)] 2,1 = - t ∞,0 ω 2 L α [ω] + (t ∞,0 ) 2 -t ∞,0 ω ν (α) ∞,-1 + t ∞,0 L α [ω] ω 2 + t ∞,0 (1 -t ∞,0 ) ω ν (α) ∞,-1 λ +O(1) = O(1) (A-94)
We still need to compute the constant term. In order to obtain it, we write entry (2, 1) of the isomonodromic compatibility equation:

L α [ L2,1 (λ)] = ∂ λ [ Ãα (λ)] 2,1 -2 L2,1 (λ)[ Ãα (λ)] 1,1 -L1,1 (λ)[ Ãα (λ)] 2,1 (A-95)
The normalization of L(λ) at infinity implies that L2,1 (λ) = O(λ -2 ). Since it is a derivative, the term ∂ λ [ Ãα (λ)] 2,1 does not provide any term or order λ -1 at infinity. In the r.h.s.

L2,1 (λ) = O(λ -2 ) while [ Ãα (λ)] 1,1 = O(1) so that L2,1 (λ)[ Ãα (λ)] 1,1 = O(λ -2 ). Let us denote A 0 the constant term at infinity of [ Ãα (λ)] 2,1 , i.e. [ Ãα (λ)] 2,1 = 2t ∞,0 ω 2 L α [ω]λ + A 0 + O(λ -1 ) (A-96) We have L1,1 (λ) = -t ∞,0 λ -1 + β -1 λ -2 + O(λ -3 ) so that order λ -1 at infinity of (A-95) provides 0 = 2t ∞,0 ω 2 L α [ω]β -1 -t ∞,0 A 0 ⇒ A 0 = 2 ω 2 L α [ω]β -1 (A-97) Finally, following Lemma A.2 we have L1,1 (λ) = L1,1 (λ) + (t ∞,0 λ + g 0 ) ω L1,2 (λ) = O(λ -3 ) (A-98) It implies that β -1 = -g 0 - t ∞,0 ω n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 (A-99)
so that we finally get

A 0 = - 2 ω 2 L α [ω] g 0 + t ∞,0 ω n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 (A-100)
Thus we end up with:

[ Ãα (λ)] 2,1 = 2t ∞,0 ω 2 L α [ω]λ - 2 ω 2 L α [ω] g 0 + t ∞,0 ω n s=1 X 2 s Q Xs,1 + 2X s Q Xs,2 + Q Xs,3 +       2rs k=rs+1 2rs j=k 2rs-j m=0 t Xs,rs-1-m t Xs,j+m-rs-1 ν (α) Xs,j-k (λ -X s ) -k n s=1 rs k=1 Q Xs,k (λ-Xs) k       Xs,- + n s=1 L1,1 (λ) L1,2 (λ) L1,1 (λ) L1,2 (λ) [ Ãα (λ)] 1,2 -2[ Ãα (λ)] 1,1 Xs,- (A-101) B Proof of Lemma 4.1 From Theorem 4.1, we introduce R Xs,k = rs+1-k m=1 P Xs,m Q Xs,k+m-1 - (g 0 + t ∞,r∞-1 X s ) ω Q Xs,k - t ∞,r∞-1 ω Q Xs,k+1 δ k≤rs-1 , ∀ (s, k) ∈ 1, n × 1, r s R ∞,r∞-4 = -ω P ∞,0 - g 0 ω - t ∞,r∞-1 ω Q ∞,r∞-4 if r ∞ ≥ 4 R ∞,k = -ω P ∞,r∞-4-k - r∞-5-k m=0 P ∞,m Q ∞,k+1+m - t ∞,r∞-1 ω Q ∞,k-1 - g 0 ω Q ∞,k , ∀ k ∈ 1, r ∞ -5 R ∞,0 = -ω P ∞,r∞-4 - r∞-5 m=0 P ∞,m Q ∞,m+1 - g 0 ω Q ∞,0 - t ∞,r∞-1 ω n s=1 Q Xs,1 (B-1) so that L1,1 (λ) λ→Xs = rs k=1 R Xs,k (λ -X s ) k + O(1) , ∀ s ∈ 1, n (B-2)
and for r ∞ ≥ 3:

L1,1 (λ) λ→∞ = -t ∞,r∞-1 λ r∞-2 -t ∞,r∞-2 λ r∞-3 + r∞-4 k=0 R ∞,k λ k + O(λ -1 ) (B-3)
Since L1,1 (λ) is a rational function of λ with poles only in {∞, X 1 , . . . , X n }, we obtain Lemma 4.1. The additional constraints for r ∞ ≤ 2 follow from the first two leading orders at infinity of L1,1 (λ).

Conditions (D-1) are equivalent to We first observe that a solution is trivially given by Q Xs,k = t Xs,k-1 u Xs,rs for all k ∈ 2, r s where u Xs,rs-1 is independent of the irregular times. It is also obvious from the lower triangular form that for any k ∈ 2, r s , Q Xs,k may only depend on t Xs,rs-1 , . . . , t i.e.

-     δ (α) t [Q Xs,rs ] rs-1 . . . δ (α) t [Q Xs,2 ] 1     =             Q Xs,
0 Q Xs,2 Q Xs,3 . . . Q Xs,rs-1 Q Xs,rs                 ν (α) Xs,1 . . . ν (
Q Xs,rs-2 = t ∞,r∞-3 u Xs,rs + r s -3 r s -2 (t Xs,rs-1 ) -1 rs-1 t Xs,rs-2 u Xs,rs-1 + (t Xs,rs-1 )

rs-3 rs-1 u Xs,rs-2

(D-11) with u Xs,rs-2 independent of the irregular times. One can then proceed by induction to obtain the next lines. In particular if we take j 0 ∈ 1, r s -1 and look for solutions of the form f j 0 (, t Xs,j 0 , . . . , t Xs,rs-1 ) = (0 j-1 , [f j 0 ] j 0 +1 (t Xs,rs-1 ), . . . , [f j 0 ] j 0 +1 (t Xs,j 0 , . . . , t Xs,rs-1 )) t . The differential system (D-5) reduces to the lower-left corner of size (r s -j 0 )×(r s -j 0 ) because all other entries are trivial. Entry (j 0 , 1) provides a simple ODE for [f Xs,j 0 ] j 0 (t Xs,rs-1 ) whose solution is provided in Theorem 5.3. Then looking at the line (j 0 +1) whose only non-trivial entries are (j 0 + 1, 1) and (j 0 + 1, 2), we obtain the explicit dependence of [f Xs,j 0 ] j 0 +1 (t Xs,rs-1 , t Xs,rs-2 ) relatively to t Xs,rs-1 ((j 0 + 1, 2) entry) and t Xs,rs-2 ((j 0 + 1, 1) entry). One then proceed with line (j 0 + 2) (whose only first three columns are non-trivial) to determine [f Xs,j 0 ] j 0 +2 (t Xs,rs-1 , t Xs,rs-2 , t Xs,rs-3 ) and a simple induction gives all coefficients of the vector f Xs,j 0 . Thus, we end up with r∞-2,r∞-4 (t ∞,r∞-3 ) 0 1 For r ∞ = 2 we may choose the variables (v Xs,1 ) 1≤s≤n so that n s=1 v Xs,1 = -t ∞,0 since -t ∞,0 does not depend on the irregular times. For r ∞ = 1, the same remark applies too. For r ∞ = 1, the second additional constraint (5-47) is also compatible and from (A-99) it provides the explicit dependence of β -1 in terms of the irregular times in order to satisfy the isospectral condition.

                      1 u ∞,r∞
The situation is a little more complicated at infinity. Let us first observe that the study at infinity is only needed for r ∞ ≥ 4 and in this case, we have c ∞,0 = 1 2ω L α [ω] = 0 from Section A.2.2 and Proposition 5.2. Thus, the isospectral condition reduces for r ∞ ≥ 4 to

[ Ãα (λ)] 1,1 = L1,1 (λ)[A α (λ)] 1,2 (E-3)
This relation is analogous to [ Ãα (λ)] 1,2 = L1,2 (λ)[A α (λ)] 1,2 that has been used to derive Proposition 5.2. The only difference is that L1,1 (λ) = -λ r∞-2 + O(λ r∞-4 ) while L1,2 (λ) = ωλ r∞-3 + O(λ r∞-4 ). However, it is obvious that the computations made for (5-28) can easily be adapted to this case. Using Lemma 4.1, equation (5-28) is adapted into

r∞-4 k=0 δ (α) t [R ∞,k ]λ k = -(r ∞ -3)ν (α)
∞,1 λ r∞-4 δ r∞≥4 -(r ∞ -4)ν Indeed, for r ∞ ≥ 3, we have t ∞,r∞-1 = 1 and t ∞,r∞-2 = 0 and there is no deformation relatively to t ∞,r∞-1 nor t ∞,r∞-2 (i.e. α ∞,r∞-1 = α ∞,r∞-2 = 0). Using Proposition 3.3, this system may be rewritten as a (r ∞ -3) × (r ∞ -3) system: It is obvious that R ∞,k = -t ∞,k+1 for all k ∈ 0, r ∞ -4 provides a solution of the system. Straightforward computations similar to Section D.2 provide the expressions presented in Theorem 5.4.

          1

Proposition 3 . 3 (

 33 Expression of ν (α) p,k (Proposition 4.1 of [24])). Let us define

Definition 4 . 3 .

 43 Let us define the coordinates R := {R ∞,0 , . . . , R ∞,r∞-3 }∪ n s=1 {R Xs,1 , . . . , R Xs,rs } by:

2 , 1

 21 is determined by[START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF][START_REF] Chernyakov | Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems[END_REF][START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Fuchs | Sur quelques équations différentielles linéaires du second ordre[END_REF],[START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF][START_REF] Chernyakov | Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems[END_REF][START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Fuchs | Sur quelques équations différentielles linéaires du second ordre[END_REF][START_REF] Gaiur | Isomonodromic deformations: Confluence, reduction & quantisation[END_REF],[START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF][START_REF] Chernyakov | Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems[END_REF][START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Fuchs | Sur quelques équations différentielles linéaires du second ordre[END_REF][START_REF] Gaiur | Isomonodromic deformations: Confluence, reduction & quantisation[END_REF][START_REF] Gambier | Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes[END_REF] or (4-17) depending on the value of r ∞ .

2 λ→Xs= 1 L1, 2 λ→Xs=

 212 O((λ-X s ) -rs ) and L1,O(1). Thus, coefficients of order (λ-X s )-(rs+1)

δt

  u ∞,k ) 0≤k≤r∞-4 independent of the irregular times. The first extra line appears because of our choice of normalization at infinity ( L1,2 = ω λ r∞-3 + O(λ r∞-4 )) that freezes a possible coordinate Q ∞,r∞-3 to ω. It is also easy to observe from the differential system (D-20) thatf (∞) k,k-2 (t ∞,r∞-3 ) = r ∞ -1 -k r ∞ -3 t ∞,r∞-3 , ∀ k ∈ 3, r ∞ -2 (D-28)E Proof of Theorem 5.4We recall that from (A-44), we have[ Ãα (λ)] 1,1 = c ∞,0 + L1,1 (λ)[A α (λ)] 1,2 (E-1)Thus, using Lemma 4.1, the isospectral condition δ(α) t [ L1,1 (λ)] = ∂ λ [ Ãα (λ)] 1,1 is identical at finite poles to the case δ (α) t [ L1,2 (λ)] = ∂ λ [ Ãα (λ)] 1,2given by (A-2). We end up with δ (α) t R Xs,1 = 0 for all s ∈ 1, n and Xs,rs-1 [RX

  ∞,r∞ -3 [R∞,r ∞ -4] . . . δt ∞,2 [R∞,r ∞ -4] . . . . . . . . . δt ∞,r∞-3 [R∞,0] . . . δt ∞,2 [R∞,0] ∞,2 R∞,r ∞ ,3 . . . . . . R∞,r ∞-

  and, for all s ∈ 1, n , H Xs = (H Xs,1 , . . . , H Xs,rs ) t . Matrices V ∞ and (V Xs ) 1≤s≤n are defined by

			1	1	. . . . . .	1
	V∞ =	       	q1 . . . . . .	q2	. . . . . .	qg . . . . . .

  We now observe that the lower triangular Toeplitz matrices commute so that conditions (D-1) are equivalent toWe may now specify the last equation for any deformation t Xs,i with i ∈ 1, r s -1 . It gives: δt Xs,rs-1 [QX s,rs ] . . . δt Xs,1 [QX s ,rs ] . . . . . . . . . δt Xs,rs -1 [QX s,2 ] . . . δt Xs ,1 [QX s ,2]

													
													 	(D-2)
													α)	
													Xs,rs-1
	i.e. using Proposition 3.3:
												Q Xs,rs	0	. . .	. . .	0	
	   	δ δ (α) t [Q Xs,rs ] rs-1 . . . (α) t [Q Xs,2 ] 1	    =	          	Q Xs,rs-1 Q Xs,rs 0 . . . . . . . . . . . . . . . . . . Q Xs,3 . . . . . .	. . . . . . . . . Q Xs,rs 0	. . . . . . . . . 0	          	(M Xs ) -1	  	α Xs,rs-1 1 rs-1 . . . α Xs,1	  
												Q Xs,2	Q Xs,3 . . .	Q Xs,rs-1 Q Xs,rs
													(D-3)
													Q Xs,rs	0	. . .	. . .	0	
	M Xs	   	δ δ (α) t [Q Xs,rs ] rs-1 . . . (α) t [Q Xs,2 ] 1	    =	           Q Xs,rs-1 Q Xs,rs 0 . . . . . . . . . . . . . . . . . . Q Xs,3 . . . . . .	. . . . . . . . . Q Xs,rs 0	. . . . . . . . . 0	          	  	α Xs,rs-1 1 rs-1 . . . α Xs,1	   (D-4)
													Q Xs,2	Q Xs,3 . . .	Q Xs,rs-1 Q Xs,rs
		     tX s,rs -1 tX s,rs -2 tX s ,rs-1 0 . . . . . . tX s,1 . . .	. . . . . . tX s,rs-2 tX s ,rs-1 0 0 . . .	    	      	1 rs-1 0 . . . 0	0 . . . . . .	. . . 0 0 1 0 1 . . . 0	      	  	  
				QX s,rs		0			. . .	. . .	0
		=	  QX s,rs -1 QX s,rs    . . . . . .    . . . . . .     QX s,3 . . .		0 . . . . . . . . .	. . . . . . . . .	0 QX s,rs	. . . . . . . . . 0
					QX s,2	QX

s ,3 . . . QX s ,rs-1 QX s ,rs

  Xs,k-1 . The first line of (D-5) provides t Xs,rs-1 δ t Xs,k Q Xs,rs r s -1 = δ k,rs-1 Q Xs,rs r s -1 (D-6) i.e. Q Xs,rs = t Xs,rs-1 u Xs,rs (D-7)with u Xs,rs independent of the irregular times. The second line of (D-5) providest Xs,rs-2 δ t Xs,rs-1 Q Xs,rs r s -1 + t Xs,rs-1 δ t Xs,rs-1 Q Xs,rs-1 r s -2 = Q Xs,rs-1 r s -1 t Xs,rs-1 δ t Xs,rs-2 Q Xs,rs-1 r s -2 = Q Xs,rs r s -2 δ t Xs,rs-k [Q Xs,rs-1 ] = 0 , ∀ k ∈ 3, r s -1 (D-8) i.e.Q Xs,rs-1 = t Xs,rs-2 u Xs,rs + (t Xs,rs-1 ) Xs,rs-1 independent of the irregular times. The third line of (D-5) providesQ Xs,rs-2 r s -1 = t Xs,rs-3 δ t Xs,rs -1 Q Xs,rs r s -1 + t Xs,rs-2 δ t Xs ,rs-1 Q Xs,rs-1 r s -2 + t Xs,rs-1 δ t Xs ,rs-1 Q Xs,rs-2 r s -3 Q Xs,rs-1 r s -2 = t Xs,rs-2 δ t Xs,rs -2 Q Xs,rs-1 r s -2 + t Xs,rs-1 δ t Xs,rs-2 Q Xs,rs-2 r s -3 Q Xs,rsr s -3 = t Xs,rs-1 δ t Xs,rs -3 Q Xs,rs-2 r s -3 0 = δ t Xs,rs -k [Q Xs,rs-1 ] , ∀ k ∈ 4, r s -1 (D-10)

	rs-2
	rs-1 u Xs,rs-1	(D-9)
	with u

  (tX s,rs -2, tX s,rs-1 ) f (tX s,1 , . . . , tX s ,rs-1) (t ∞,r∞-4 , t ∞,r∞-3 ) f

									1	0	0	. . .	. . . 0
									0	1	0	. . .	. . .
								f	(∞) 3,1 (t ∞,r∞-3 )	0	1	0	0
						     f	f (∞) (∞) 4,1 (∞) 4,2 (t ∞,r∞-3 ) 0 . . . . . .	1	. . . . . . 0 . . .
		QX s ,rs						f 1,1 (tX s,rs -1) (Xs)	0	. . .	. . .	0			uX s,rs	
	      	QX s,rs-1 . . . . . .	      	=	       	f	(Xs) 2,1 (Xs) 2,2 (tX s,rs-1 ) . . .	0 . . .	. . .	. . . . . .	       	       uX s ,rs-1 . . . . . .	      
		QX s ,2				(Xs) rs-1,1 . . .		. . .	f	uX s,2
													(D-12)

f (Xs) rs-2,1 (tX s ,2, . . . , tX s ,rs-1) . . . . . . f (Xs) rs-2,rs-1 (tX s ,rs-1) 0 f (Xs) rs-1,rs-1 (tX s,rs -1)

with all (u Xs,k ) 1≤k≤rs independent of the irregular times. Moreover, we have:

f (Xs) j,j (t Xs,rs-1 ) = (t Xs,rs-1 ) rs-j rs-1 , ∀ j ∈ 1, r s -1 r∞-2,1 (t ∞,2 , . . . , t ∞,r∞-3 ) . . . f (∞)

  s,rs ] . . . δt Xs,1 [RX s,rs ] . . . . . . . . . δt Xs,rs -1 [RX s,2 ] . . . δt Xs,1 [RX s,2 ]

							
							
							
			RX s ,rs	0	. . .		. . .	0
	=	  RX s,rs -1 RX s,rs    . . . . . .    . . . . . .     RX s,3 . . .	0 . . . . . . . . .	. . . . . . . . .	0 RX s,rs	. . . . . . . . . 0
			RX s,2

RX s,3 . . . RX s,rs-1 RX s ,rs

In[START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF], ω was set to 1 for simplicity since the Hamiltonian system does not depend on the choice of ω. However, as we will see it in Section 5, imposing the isospectral condition of[START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF] requires that ω must depend on the irregular times when r∞ = 1 so that in this context, one cannot simply set it to 1.

The derivation of c∞,0 was not done in[START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF] because it is irrelevant in the computation of the Hamiltonian system but only related to the choice of normalization of L. For completeness, the formula is proved in Appendix A.2.2.

Acknowledgements

The authors would like to thank John Harnad for fruitful discussions on the isospectral condition. The authors would also like to thank Nicolas Orantin for his support at the early stage of this project.

Since we know that [ Ãα (λ)] 1,2 is a rational function of λ with only poles at λ ∈ {X 1 , . . . , X n , ∞} we end up with Ãα (λ)

and (A-4) and (A-5) implies that we get some extra conditions for r ∞ = 2 and r ∞ = 1:

• For r ∞ = 2:

• For r ∞ = 1: A.2 Entry (1, 1)

Let us start with the following lemma:

Lemma A.1. We have for all i ∈ 1, g :

Proof. Let i ∈ 1, g . Taking the derivative relatively to q i in the definition of the change of coordinates of Definition 4.1 provides

A.2. For r ∞ = 2, the l.h.s. of (A-17) is of order λ -2 so that we need to impose the condition n s=1 rs m=1 P Xs,m Q Xs,m = 0 so that the r.h.s. has the same property. Similarly for r ∞ = 1, the l.h.s.

of (A-17) is of order λ -3 so that we need to impose the conditions X s P Xs,m Q Xs,m = 0 so that the r.h.s. has the same property.

Finally, the gauge transformation (3-1) implies that

Let us now look at the behavior at infinity.

We have that

Hence we get

). We have that

) so that the r.h.s. of (A-19) is

Hence we get

). We have also from Lemma A.2:

so that using, n s=1 Q Xs,1 = ω, the r.h.s. of (A-19) is:

Hence we get

Hence, Lemma A.2 provides the expression for Theorem 4.1.

Let us now observe that entry (2, 1) of the compatibility equations implies:

From the gauge transformation we have

1,2 (λ) so that we end up with

From [START_REF] Adams | Dual moment maps into loop algebras[END_REF][START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF][START_REF] Chernyakov | Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems[END_REF][START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Fuchs | Sur quelques équations différentielles linéaires du second ordre[END_REF][START_REF] Gaiur | Isomonodromic deformations: Confluence, reduction & quantisation[END_REF][START_REF] Gambier | Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes[END_REF][START_REF] Garnier | Solution du problème de Riemann pour les systèmes différentiels linéaires du second ordre[END_REF][START_REF] Harnad | Dual isomonodromic deformations and moment maps to loop algebras[END_REF][START_REF] Harnad | Hamiltonian Dynamics, Classical R-matrices and Isomonodromic Deformations[END_REF], we have

). Let us now discuss the various cases:

Identifying coefficient λ r∞-3 at infinity provides

Identifying coefficient λ 0 at infinity provides

Entry (1, 1) of the gauge transformation (A-1) implies that

From [START_REF] Marchal | Hamiltonian representation of isomonodromic deformations of general rational connections on[END_REF], [A α (λ)] 1,1 is given by [START_REF] Adams | Dual moment maps into loop algebras[END_REF][START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF][START_REF] Chernyakov | Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems[END_REF][START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Fuchs | Sur quelques équations différentielles linéaires du second ordre[END_REF][START_REF] Gaiur | Isomonodromic deformations: Confluence, reduction & quantisation[END_REF][START_REF] Gambier | Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes[END_REF][START_REF] Garnier | Solution du problème de Riemann pour les systèmes différentiels linéaires du second ordre[END_REF][START_REF] Harnad | Dual isomonodromic deformations and moment maps to loop algebras[END_REF][START_REF] Harnad | Hamiltonian Dynamics, Classical R-matrices and Isomonodromic Deformations[END_REF]. Since we know that [ Ãα (λ)] 1,1 is a rational function of λ with poles in {X 1 , . . . , X n , ∞}, expression [START_REF] Adams | Dual moment maps into loop algebras[END_REF][START_REF] Babujian | Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping[END_REF][START_REF] Ben-Zvi | Spectral curves, opers and integrable systems[END_REF][START_REF] Bertola | Hamiltonian structure of rational isomonodromic deformation systems[END_REF][START_REF] Bertola | Tau-functions and monodromy symplectomorphisms[END_REF][START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF][START_REF] Boalch | Simply-laced isomonodromy systems[END_REF][START_REF] Boalch | Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids[END_REF][START_REF] Chekhov | Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras[END_REF][START_REF] Chernyakov | Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems[END_REF][START_REF] Chiba | Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation[END_REF][START_REF] Fuchs | Sur quelques équations différentielles linéaires du second ordre[END_REF][START_REF] Gaiur | Isomonodromic deformations: Confluence, reduction & quantisation[END_REF][START_REF] Gambier | Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes[END_REF][START_REF] Garnier | Solution du problème de Riemann pour les systèmes différentiels linéaires du second ordre[END_REF][START_REF] Harnad | Dual isomonodromic deformations and moment maps to loop algebras[END_REF][START_REF] Harnad | Hamiltonian Dynamics, Classical R-matrices and Isomonodromic Deformations[END_REF] implies that [A α (λ)] 1,1 only contributes with a factor c ∞,0 . Hence, (A-43) reduces to

We now need to evaluate the r.h.s. at each pole. We have for any s ∈ 1, n :

Xs,0 (λ -Xs) -rs

At infinity, for r ∞ ≥ 3, we have:

Note that for r ∞ ≥ 3, the term in λ 0 implies ν

∞,r∞-2 . In order to determine it, we simply look at order λ -1 at λ → ∞ of [ Ãα (λ)] 1,2 in Theorem 4.1. We find:

For r ∞ ≥ 3 :

Since, we shall need it later, one may also determine ν

∞,r∞-1 by looking at order λ -2 at λ → ∞ of [ Ãα (λ)] 1,2 in Theorem 4.1 and we find

Thus we end up with the formula given by Theorem 4.1.

A.3 Entry (2, 1)

The gauge transformation (3-1) implies that

Entry (2, 1) of the previous equality provides

L1,2 (λ) . Thus, we end up with

• If r ∞ = 1, we have from the definition of L 2,1 (λ) :

Thus we get:

Let us now study the term L1,1 (λ) L1,2 (λ)

As mentioned in (A-77) we have:

Regrouping results

Let us regroup all the contributions in each case.

C Proof of Theorem 4.3

We have from (A-44)

Contributions at infinity for r ∞ ≤ 2 are similar to (A-49) using expression of c ∞,0 of Section A.2.2. For any s ∈ 1, n , we have

At infinity, for r ∞ ≥ 3, we have

Thus, we end up with formulas of Theorem 4.3.

D Proof of Theorem 5.3 D.1 Study at a finite pole

From Proposition 5.2, the condition to satisfy is

Let us now discuss the special cases of r ∞ = 2 and r ∞ = 1.

• For r ∞ = 2, we need to impose the extra condition

This is equivalent to

t

[ω] = 0 so that ω does not depend on the trivial times.

• For r ∞ = 1, we need to impose the extra conditions

u Xs,1 = 0 which is obviously consistent. On the contrary, the second one does not look consistent a priori because Q Xs,2 may depend on the irregular times in a very non-trivial way. However, for r ∞ = 1, the additional constraints (4-9) provide

Extra conditions (4-3) in Definition 4.1 imply that the l.h.s. is δ

D.2 Study at infinity

The situation at infinity is a little more complicated due to the normalization of the Lax matrices. Let us take r ∞ ≥ 5 which is the only non-trivial case. From Proposition 5.2, the condition to satisfy implies Q ∞,r∞-4 = ω u ∞,r∞-4 and ω independent of the irregular times. Moreover, we have:

This can be rewritten into a (r ∞ -4) × (r ∞ -4) linear system:

Using the first (r ∞ -4) lines of Proposition 3.3, we get

Note that the r.h.s. does not involve α ∞,1 nor t ∞,1 so that all (Q ∞,k ) 0≤k≤r∞-4 are independent of t ∞,1 . Specializing δ (α) t

to each δ t ∞,i with i ∈ 2, r ∞ -3 and using the commutation of lowertriangular Toeplitz matrices and the fact that t ∞,r∞-1 = 1 and t ∞,r∞-2 = 0 (because r ∞ ≥ 3), we end up with :

. . . 0 0 The first line of (D-20) provides

with u ∞,r∞-5 independent of the irregular times. The second line of (D-20) provides

with u ∞,r∞-6 independent of the irregular times. The third line of (D-20) provides

i.e.

Q ∞,r∞-7 = (r ∞ -6) (r ∞ -5) ω t ∞,r∞-5 + (r ∞ -6) (r ∞ -4) ω t ∞,r∞-4 u ∞,r∞-4 + (r ∞ -4)(r ∞ -6)ω 2(r ∞ -5)(r ∞ -3) (t ∞,r∞-3 ) 2 + (r ∞ -6) (r ∞ -3) t ∞,r∞-3 ω u ∞,r∞-5 -(r ∞ -6)(r ∞ -4)ω 2(r ∞ -5) (t ∞,r∞-3 ) 2 + ω u ∞,r∞-7