Notes on Non-Generic Isomonodromy Deformations

Some of the main results of [Cotti G., Dubrovin B., Guzzetti D., Duke Math. J., to appear, arXiv:1706.04808], concerning non-generic isomonodromy deformations of a certain linear differential system with irregular singularity and coalescing eigenvalues, are reviewed from the point of view of Pfaffian systems, making a distinction between weak and strong isomonodromic deformations. Such distinction has a counterpart in the case of Fuchsian systems, which is well known as Schlesinger and non-Schlesinger deformations, reviewed in Appendix A.


Introduction
These notes partly touch the topics of my talk in Ann Arbor at the conference in memory of Andrei Kapaev, August 2017. They are a reworking of some of the main results of [12], concerning non-generic isomonodromy deformations of the differential system (1.1) below. The approach here is different from [12], since I will start from the point of view of Pfaffian systems. This allows to introduce the main theorem (Theorem 3.2 below) in a relatively simple way (provided we give for granted another result of [12] summarised in Theorem 3.1 below). The approach here is also an opportunity to review the difference between weak and strong isomonodromy deformations.
In Section 1, we define "weak" isomonodromic deformations of the system (1.1), through the Pfaffian system (1.2) below, that we characterise together with its fundamental matrix solutions. The residue matrix Apuq is not assumed to be diagonalizable or non-resonant, so that a first non-generic issue is included in the discussion. In Section 2, we define and characterise "strong" isomonodromy deformations of the differential system (1.1) in terms of solutions of the Pfaffian system and in terms of essential monodromy data. We recover the total differential system used in the last part of [12], a special case of which (for Apuq skewsymmetric) has been well known in the theory of Frobenius manifolds [16], [18]. In Section 3, we explain some of the main results of [12], particularly Theorem 3.2, which extend strong isomonodromy deformations to the non-generic case when the matrix Λ " diagpu 1 , ..., u n q in system (1.1) has coalescing eigenvalues pu i´uj q Ñ 0 for some 1 ď i ‰ j ď n. The proof of Theorem 3.2 is based on Theorem 3.1, which holds for a differential system more general than (1.1), not necessarily isomonodromic (see system (3.7)). Paper [12] is unavoidably long because of a careful set up for the background and the proof of Theorem 3.1. Given it for granted, in these notes we can introduce and prove Theorem 3.2 in a relatively short manner, starting from the discussion of the Pfaffian system (1.2).
Weak isomonodromy deformations are the natural framework for isomonodromic Fuchsian systems; in this case, "weak" and "strong" deformations respectively yield the non-Schlesinger and the Schlesinger deformations. We will review this issue in Appendix I.
Several detailed comments on the existing literature concerning non-generic isomonodromy deformations have been included in the introduction of [12], so we do not repeated them here. However, a comment is missing from the cited introduction, which I will include here in Remark 3.2 of Section 3 below.

A Pfaffian System defining Weak Isomonodromic Deformations
In this section, particularly in Proposition 1.2, we characterise a Pfaffian system responsible for the (weak) isomonodromic deformations of the differential system B z Y "ˆΛ`A puq z˙Y . (1.1) The above appears naturally as the (inverse) Laplace transform of a Fuchsian system of Okubo type, with poles at u 1 , ..., u n [9], [29], [19], [22]. A particular case of (1.1) is at the core of the isomonodromic deformation approach to Frobenius manifolds [16] [18]. Consider a nˆn matrix Pfaffian system dY " ωY, ω " ω 0 pz, uqdz`n ÿ j"1 ω j pz, uqdu j . (1. 2) The complex n`1 variables will be denoted by pz, uq, with u :" pu 1 , ..., u n q. We assume that with Λ " diagpu 1 , ..., u n q and Apuq a nˆn matrix, so that (1.2) can be viewed as a deformation of the differential system (1.1). We suppose that ω 1 , ..., ω n are holomorphic of pz, uq P CˆDpu 0 q, where Dpu 0 q is a polydisc centred at u 0 " pu 0 1 , ..., u 0 n q, contained in We also assume that Apuq is holomorphic on Dpu 0 q and that z " 8 is at most a pole of the ω j pz, uq, so that (1.2) is meromorphic on P 1ˆD pu 0 q. The complete Frobenius integrability of the system (1.2) is expressed by dω " ω^ω, namely, letting px 0 , x 1 , ..., x n q :" pz, u 1 , ..., u n q, Bω α Bx β`ω α ω β , α ‰ β " 0, 1, ..., n. (1.4) If the integrability condition holds ( [3], [34], [23], [24]), then (1.2) admits of a fundamental matrix solution Y pz, uq, which is holomorphic on RˆDpu 0 q, where R is the universal covering of P 1 zt0, 8u. Its monodromy matrix M associated with a simple loop γ around z " 0, defined by Y pγz, uq " Y pz, uqM, is independent of u (and of course of z). The notation above means that γ transforms z P R to γz P R (z and γz are in the same fibre over a point of P 1 zt0, 8u, which we still denote by z, to avoid heavy notations). To show that M is constant, observe that Y pγz, uq is a solution (it is Y p¨, uq seen as function on R evaluated at γz). Therefore, ω " dY pγz, uq¨Y pγz, uq´1 " dY pz, uq¨Y pz, uq´1`Y pz, uqpdM¨M´1qY pz, uq´1 (1.5) " ω`Y pz, uqpdM¨M´1qY pz, uq´1 ðñ dM " 0. (1.6) In order to avoid heavy notations, we use the letter "d" either for the differential of a function f pz, uq of variables pz, uq (like dY above), and for the differential of a function of u alone (like dM above).
Definition 1.1. We call (1.1) a weak isomonodromic family of differential systems, and Y pz, uq a weak isomonodromic family of fundamental matrix solutions.
The last expression is zero for any possible A if and only if (1.10) holds. l The following is a standard result concerning the residue matrix at Fuchsian singularity of a Pfaffian system (see for example [23], [34], [3]). Proposition 1.1. Let ω 0 be as in (1.3), let ω 1 , ..., ω n be holomorphic of pz, uq P CˆDpu 0 q, and Apuq holomorphic on Dpu 0 q. Then, Apuq is holomorphically similar to a constant (i.e. independent of u) Jordan form J, namely there exists a holomorphic fundamental matrix Gpuq of (1.8) on Dpu 0 q, such that Gpuq´1ApuqGpuq " J.
In particular, this means that the weak isomonodromy deformation (1.1) is isospectral.
Proof: Take a fundamental matrixGpuq of (1.8), which is holomorphic on Dpu 0 q. Then, using (1.7) and (1.8), we have G´1A ω j p0, uqG`G´1B j AG´G´1ω j p0, uqAG "G´1 prA, ω j p0, uqs`B j AqG " 0. This implies that A :"G´1AG is constant over Dpu 0 q, and it has a constant Jordan form J " G´1AG for some invertible matrix G. Thus, the desired diagonalising holomorphic matrix is Gpuq "GpuqG. l At any fixed u, we can take a fundamental matrix of the differential system (1.1) is a convergent matrix valued Taylor series (I stands for the identity matrix). The matrix Gpuq puts Apuq in Jordan form J " Gpuq´1ApuqGpuq and satisfies (1.8) as in Proposition 1.1. The following characterisation holds [21], [1].
-Lpuq is block-diagonal L " L 1 '¨¨¨' L , with upper triangular matrices L q ; each L q has only one eigenvalue σ q , satisfying 0 ď eσ q ă 1, and σ p ‰ σ q for 1 ď p ‰ q ď . -D is a diagonal matrix of integers, which can be split into blocks D 1 '¨¨¨' D as L. The integers d q,r in each D q " diagpd q,1 , d q,2 , ....q form a non-increasing (finite) sequence d q,1 ě d q,2 ě .... -The eigenvalues of A are d q,s`σq , for q " 1, ..., and s runs from 1 to some integer n q , if the dimension of L q is n qˆnq . Each block L q puq corresponds to the eigenvalues of Apuq which differ by non-zero integers. -The expression in square brackets below is holomorphic at z " 0 and the following limits hold The matrices Ψ j puq, Lpuq are not necessarily holomorphic, and the integer diagonal matrix D may be discontinuous with respect to u. We also recall that a solution (1.11) is not uniquely determined by the differential system, since there is a freedom in the choice of the coefficients Ψ j puq and the exponents D and L. We will not discuss this freedom here, since we will not use it. See [12] (and also [18]) for more details. Remark 1.1. Notice that we can write L " Σ`N , where Σ " Σ 1 '¨¨¨'Σ is diagonal, with Σ q " σ q I dimpΣqq (here I dimpΣqq stands for the identity matrix of dimension dimpΣ q q), while N " N 1 '¨¨¨' N is nilpotent. Since Σ q and N q have the same dimension (q " 1, ..., ), we have rΣ, N s " 0, Given the holomorphic isomonodromic family Y pz, uq, at each u there exists a connection matrix Hpuq such that where L :" Hpuq´1LpuqHpuq. Clearly, so that L is constant. Proof: Since dM " 0, then dL " 0, and thus the eigenvalues σ q of L are independent of u. Moreover, the eigenvalues µ q,s puq " σ q`dq,s puq of Apuq are continuous, being Apuq continuous, so that all the integer d q,s must be constant. Now, Gpuq is holomorphic by Proposition 1.1, D and the eigenvalues of A are constant. Hence, the recursive procedure which explicitly constructs a Levelt form [21], [33] allows to choose each Ψ j puq and Lpuq holomorphic on the domain where Apuq is holomorphic. l The system (1.1) also admits of family of fundamental matrices Y r pz, uq , r P Z, uniquely determined by their asymptotic behaviour in suitable sectors of central angular opening greater that π. We fix a half-line of direction arg z " τ in R which does not coincide with any of the Stokes rays associated with Λpu 0 q, namely with the half lines in R specified by 1 pzpu 0 i´u 0 j qq " 0. We say that τ is an admissible direction at u 0 . Notice that we are working in the universal covering R, so that any τ`hπ, h P Z is also an admissible direction. We define the Stokes rays associated with Λpuq to be the infinitely many half-lines in R defined by pzpu i´uj qq " 0 These rays rotate as u varies in Dpu 0 q, and may cross an admissible direction. Let D 1 Ă Dpu 0 q be a disk sufficiently small so that no Stokes rays cross a direction τ`hπ as u varies in the closure of D 1 . Then, for u P D 1 , we construct a sector S 1 puq, by considering the "half plane" τ´π ă arg z ă τ and extending it to the nearest Stokes rays associated with Λpuq lying outside the half plane. Analogously, for any r P Z a sector S r puq is obtained extending the "half plane" τ´pr´2qπ ă arg z ă τ`pr´1qπ to the nearest Stokes rays outside of it. The sectors S r puq have central angular opening greater than π, and the same holds for S r pD 1 q :" (1.16) Since u i ‰ u j for i ‰ j, it is well known that the system (1.1) has a formal fundamental matrix solution where Bpuq " diagpA 11 puq, ..., A nn puqq is diagonal, and F pz, uq is the formal matrix-valued expansion F pz, uq " I`8 with holomorphic on Dpu 0 q coefficients F k puq uniquely determined by (1.1). Moreover, for each fixed u, and for any sector S containing in its interior a set of basic Stokes rays 2 and no other Stokes rays, it is well known [6] that there exists a unique fundamental matrix solution Now, if u is restricted to D 1 , then we can take a family of such solutions, labelled by r P Z, with S " S r puq, In order to obtain half-lines, we also need to specify the sign the imaginary part, such as pzpu 0 i´u 0 j qq ă 0. The same observation holds for the Stokes rays pzpu i´uj qq " 0 introduced below, which are specified also by the condition pzpu iú j qq ă 0. 2 Let µ be an integer. We say that a finite sequence of Stokes rays arg z " τ 1 , arg z " τ 2 ,¨¨¨, arg z " τµ, τ 1 ă τ 2 ă¨¨¨ă τµ, pτµ´τ 1 q ă π, form a set of basic Stokes rays if all the other Stokes rays can be obtained as arg z " τν`kπ, for some ν P t1, 2, ..., µu and k P Z.
There exist a set of basic rays (and then infinitely many). Indeed, a sector with angular opening π, whose boundary rays are not Stokes rays, contains exactly a set of basic rays.
It is a fundamental result of Sibuya's [30], [31], [32] that each Y r pz, uq depends holomorphically on u P D 1 , and the asymptotics (1.19) is uniform for z Ñ 8 in S r pD 1 q with respect to u varying in D 1 . We notice that it may be necessary to further restrict D 1 , because Sibuya's result requires D 1 to be sufficiently small in order for the holomorphic dependence to occur. The Stokes matrices S r defined by are holomorphic on D 1 . We recall that the structure of the Stokes matrices is such that diagpS r q " I, and pS r q ij " 0 for i, j such that e pui´uj qz Ñ 8, z P S r pD 1 q X S r`1 pD 1 q.
This is a "triangular structure". Successive matrices S r and S r`1 have opposite "triangular" structures. Given the holomorphic isomonodromic family Y pz, uq, there will exist holomorphic connection matrices H r puq, u P D 1 , such that Y pz, uq " Y r pz, uqH r puq, u P D 1 .
(1.20) Let E j be the matrix with the only non-zero entry being pE j q jj " 1. Notice that for distinct eigenvalues we have B j Λ " E j . The following proposition holds.

Proposition 1.2. Consider a completely integrable linear Pfaffian system
with Apuq holomorphic on Dpu 0 q and ω 1 , ..., ω n holomorphic of pz, uq P CˆDpu 0 q. Let z " 8 be at most a pole of ω 1 , ..., ω n . Then: (A) Each ω j pz, uq is linear in z and determined by Apuq up to an arbitrary holomorphic diagonal matrix D j puq on Dpu 0 q, with the following structure 3 ω j pz, uq " zE j`ωj p0, uq, ω j p0, uq :"ˆA ab pδ aj´δbj q u a´ub˙n a,b"1`D j puq, where D j puq is obtained by differentiating a matrix D on Dpu 0 q, whose diagonal entries only depend on u: (B) Apuq satisfies (1.7): BA Bu j " rω j p0, uq, As, j " 1, ..., n; (C) the above non where P pz, uq is polynomial in z with no constant term and with coefficients depending holomorphically on u P Dpu 0 q.
We prove the proposition in Appendix II. We only notice here that by substitution of (1.18) and (1.17) into (1.1), we find Therefore (1.23) Remark 1.2. In this paper, we have assumed that each ω j pz, uq is holomorphic at z " 0. More generally, one may study a Pfaffian system with Laurent expansions at z " 0 where ω p0q puq " ωp0, uq. Here, we only remark that if Apuq is non-resonant for each u in the domain of interest (it means that the difference of two eigenvalues of Apuq cannot be a non-zero integer), then the Frobenius integrability conditions imply that ω j pz, uq is holomorphic at z " 0, namely To see this, substitute the Laurent expansion into (1.9). We find the following recurrence relations.
-From the negative powers of z: pA`mqω p´mq´ωp´mq A " rω p´m´1q , Λs, m " p j´1 , p j´2 , ..., 1 p2q and BA Bu j " rω p0q , As`rω p´1q , Λs. p3q -From the power z 0 : If A is non-resonant, the Sylvester equation (1) determines ω p´pj q j " 0, thus the Sylvester equations (2) yield ω Finally, we notice that, independently of the resonance properties of A, equation (4) determines the off-diagonal entries of ω p0q j in terms of ω p1q j , to be pω p0q j q ab " pu a´ub q´1prω p1q j , As`ω p1q j q ab . Moreover, the diagonal part of (4) yields pω p1q j q aa " δ ja´ř b‰a ppω p1q j q ab A ba´Aab pω p1q j q ba q. Under the assumptions of Proposition 1.2, we find that ω p1q j " E j and ω pm`1q j " 0 for m ě 1. l

Strong Isomonodromic Deformations
In this section, we define strong isomonodromy deformations of the differential system (1.1), which preserve e set of monodromy data, we characterise the Pfaffian system responsible for them and its fundamental matrix solutions. The assumption here is that Λ has distinct eigenvalues, namely we work on the domain Dpu 0 q previously introduced. In the next section, we will drop this assumption.
The definition of isomonodromic deformations given by Jimbo, Miwa and Ueno in [25] is stronger that the one defined in the previous section. It requires that a set of essential monodromy data, not just the monodromy matrices, are constant. In the standard theory of [25], the matrix residue Apuq at a Fuchsian singularity must be non-resonant and reducible to a diagonal form with distinct eigenvalues, namely the Levelt form (1.11) is assumed to be Y p0q pz, uq " p Y p0q pz, uqz Λ0puq , where Λ 0 is a diagonal matrix with distinct eigenvalues. Here, we do not assume this restrictive limitation, so introducing a first non-generic feature. Definition 2.1. Let Y pz, uq be a (holomorphic on RˆDpu 0 q) fundamental matrix solution of the Pfaffian system (1.21) with coefficients (1.23). We call (1.1) a strong isomonodromic family of differential systems, and Y pz, uq a strong isomonodromic family of fundamental solutions over Dpu 0 q, if the two following conditions hold. For all r P Z the connection matrices H r in (1.20) are independent of u, namely dH r " 0; In terms of solutions of the Pfaffian system, we have the following characterisation.
It is a standard result that S 0 and S 1 , together with B, suffice to generate S r for any r P Z, through the well known formula [6] S 2r " e´2 rπiB S 0 e 2rπiB , S 2r`1 " e´2 rπiB S 1 e 2rπiB .
(2.2) We introduce connection matrices C r such that C 0 and the Stokes matrices suffice to generate all the matrices The matrices S r , B, D, L, C r (r P Z) are called the essential monodromy data of the family Y r pz, uq and Y p0q pz, uq of fundamental matrix solutions of the system (1.1). In view of (2.2) and (2.3), it suffices to consider the data S 0 , S 1 , B, D, L, C 0 . or equivalently, for some fixed value of r, The above definition is similar to the definition of monodromy data given in [25], here including the case when A may be resonant and/or non-diagonalizable. We have the following characterisation of strong isomonodromic deformations in terms of essential monodromy data.

Proposition 2.2. A deformation is strongly isomonodromic (Definition 2.1) if and only if for one value of r (and so for all) it happens that
dS r " dS r`1 " 0, dB " 0, and moreover there exists Y p0q pz, uq in Levelt form such that We observe that if the Proposition holds for S 0 , S 1 and C 0 , then it holds for any S r and C r (and conversely), by formulae (2.2) and (2.3).
Proof: First of all, we recall that for a weak isomonodromic deformation D and Σ in Remark 1.1 are constant (see Lemma 1.2), so we have nothing to prove about D.
‚ Suppose that the deformation is strong. Take Y p0q as in Definition 2.1. First, we prove that dL " 0. By Proposition 2.1 (and recalling Proposition 1.2), we have which is holomorphic at z " 0. On the other hand (recall from Remark 1.1 that L " Σ`N ), Here, by holompz, uq we mean some convergent Taylor series at z " 0, of order Op1q for z Ñ 0, with holomorphic coefficiens depending on u P Dpu 0 q. The above is holomorphic at z " 0 if and only if We prove that dC r " 0 for any r, using again Proposition 2.1: We prove that dB " 0. Actually, there is nothing to prove, because dB " 0 is already stated in Proposition 1.2. Equivalently, we can invoke Proposition 2.1. We recall that p Y r " I`F 1 z´1`Opz´2q, and we compute This can only happen if and only if dB " 0 and D j puq " 0.
We prove that dS r " 0. By Proposition 2.1, we have ‚ Conversely, we assume that dS 0 " dS 1 " dB " 0 and that there exists Y p0q such that dC 0 " dL " 0, and D constant. First, we see that dS r " 0 and dC r " 0 for any r, by formulae (2.2) and (2.3). By virtue of Proposition 2.1, it suffices to show that dY r " ωY r for any r, and that dY p0q " ωY p0q . As far as ω 0 is concerned, we already have by construction that each Y r and Y p0q satisfy the differential system (1.1), so we have nothing to prove about it.
Since dC 0 " 0 and each dS r " 0, the following 1-form is well defined and it is single-valued with respect to a loop γ around z " 0. Indeed, dL " 0 implies constancy of the monodromy matrix M p0q :" e 2πiL of Y p0q associated with γ. Moreover, the monodromy matrix of each Y r associated with γ is which is also constant, because dB " 0 and each dS r " 0. Therefore, we have " dY r pz, uq¨Y r pz, uq´1 In conclusion, each Y r and Y p0q satisfy dY " p ωY.
We can now conclude the proof invoking Proposition 2.1 if we show that p ω " ω. To this end, first we show that each ω j pz, uq is holomorphic at z " 0. Indeed, since dL " 0 and D is constant, we have where B j G¨G´1 " ω j p0, uq by Lemma 1.1. At z " 8, using the fact that dB " 0 and B j Λ " E j , we have for any r P Z The above Opz´1q stands for an asymptotic expansion, which is given by the same series in z´1 for any r, in S r pD 1 q. Since p ω j is single valued in z, then Opz´1q is a convergent Taylor series at z " 8. Thus, p ω j´z E j´r F 1 , E j s is holomorphic both at z " 0 and z " 8, and it vanishes as z Ñ 8. Hence, by Liouville theorem, which coincides with ω j with D j " 0. We have proved the equality 22)), then the equality p ω j " ω holds on Dpu 0 q. This prove the Proposition. l Proposition 2.2 says that one can give a definition of strong isomonodromic deformations alternative to Definition 2.1, namely a deformation with constant essential monodromy data on D 1 . This is the definition adopted in [25] in case Apuq is diagonalizable with distinct eigenvalues and no resonances. Here, we have worked out the general case and shown that the corresponding Pfaffian system has coefficients (2.1).

Remark 2.1.
In case Apuq is skew-symmetric and diagonalizable, the above characterisations of isomonodromic deformations with form (2.1) are well known from the work of B. Dubrovin [16], [18], where Apuq is named V puq and Λ is called U .

The non-generic case of Coalescing Eigenvalues of Λpuq
Having defined weak and strong isomonodromic deformations when the eigenvalues of Λ are distinct and Apuq is any, the next step towards non-generic isomonodromic deformations is to extended weak and strong deformations to the case when some eigenvalues of Λ coalesce, namely when u i´uj Ñ 0 for some i ‰ j. In this section, we give a "holomorphic" extension, summarised in Theorem 3.2 below, which constitutes one of the main results of [12].
Let u C " pu C 1 , ..., u C n q P C n be a coalescence point (here, "C" stands for "coalescence"), namely We consider a polydisc of radius ą 0 centered at u C and denote it by U pu C q. The coordinates of points u " pu 1 , ..., u n q in the polydisc can be represented as where t j " u j´u C j varies in the polydisc U p0q centered at t " 0. In the variables t " pt 1 , ..., t n q, then t " 0 is a coalescence point. It is to be noticed that there exists a coalescence locus in U pu C q, let it be denoted by ∆, containing u C and defined by ∆ :" U pu C q X˜ď i‰j tu i´uj " 0u¸.
In onder to study the local theory at u C , we assume that is small, so that u C is "the most" coalescent point. This means that if k ‰ l are indexes such that u C k´u C l ‰ 0, then is sufficiently small to guarantee that u k´ul ‰ 0 for every point of U pu C q.
We fix a half-line of direction arg z " r τ in R which, now, does not coincide with any of the Stokes rays associated with Λpu C q, namely with the half lines in R specified by pzpu C k´u C l qq " 0, for k, l such that u C k´u C l ‰ 0. We call r τ an admissible direction at u C . The choice of r τ determines a cell decomposition of U pu C q, which is based on two ingredients. One is ∆ above. The other one is the so called "crossing locus". In order to describe it, observe that if u P U pu C q ( points of ∆ are not excluded) is such that u i ‰ u j for some i ‰ j, then the (infinitely many in R) Stokes rays pzpu i´uj qq " 0, corresponding to pu i , u j q, are well defined. The crossing locus Xpr τ q is made of those points such that some Stokes rays "cross" the admissible rays tz P R | arg z " r τ`hπu, h P Z. Namely, Xpr τ q is made of points u such that pe ir τ pu i´uj qq " 0 for some u i ‰ u j . Precisely, u P U pu C q such that u i ‰ u j and argpu i´uj q " 3π 2´r τ mod π * Let the "walls" be defined as W pr τ q :" ∆ Y Xpr τ q. Following [12], every connected component of U pu C qzW pr τ q is called a r τ -cell. We have proved in [12] that every r τ -cell is a topological cell, so in particular it is simply connected (simple connectedness is important for the proof, given in [12], of Proposition 3.1 below).
The isomonodromy deformation theory can be extended, in a holomorphic way, to the case of coalescing eigenvalues when a certain vanishing condition holds for the entries of Apuq. i) The coefficients ω j pz, uq in (1.23) extend holomorphically on U pu C q if and only if D j puq is holomorphic on U pu C q and the following vanishing conditions hold in U pu C q: In this case, the system (1.21), (1.23) is Frobenius integrable on the whole U pu C q. ii) A fundamental matrix solution Y pz, uq exists holomorphic on RˆU pu C q if and only if (3.1) holds.
Proof: The statement follows simply by recalling that, by Proposition 1.2, for u P Dpu 0 q we have (1.23), namely so that if u i´uj Ñ 0, the condition A ij Ñ 0 guarantees analyticity on U pu C q. The integrability condition (1.4) then holds by analytic continuation from Dpu 0 q to U pu C q. The last statement concerning Y pz, uq follows from the fact that if (3.1) holds, then the Pfaffian system is integrable and linear on U pu C q with holomorphic (in u) coefficients. Conversely, if a fundamental matrix Y pz, uq exists holomorphic on RÛ pu C q, then ω " dY Y´1 has coefficients holomorphic in u on U pu C q, which implies (3.1). l The following corollary follows immediately from Proposition 1.1.
with p Y p0q pz, uq " GpuqpI`ř 8 j"1 Ψ j puqz j q, can be constructed such that Ψ j puq, Lpuq, are holomorphic on U pu C q, being D and the eigenvalues of Lpuq constant. Moreover, letting Y pz, uq " Y p0q pz, uqHpuq, then also Hpuq is holomorphic on U pu C q.
We turn to the fundamental matrices Y r pz, uq in (1.18)-(1.19). They are well defined an holomorphic on a small D 1 Ă Dpu 0 q and all the results described in Section 1 apply on D 1 . Complications arise if we want to study the matrices Y r pz, uq on the whole U pu C q.
The first delicate issue is that when u moves outside D 1 , Sibuya's local result concerning analyticity in u does no longer apply. This was analyszed in [12], for a system (1.1) without any assumption that it be isomonodromic. The following holds (indeed, for the more general system (3.7) below).

Proposition 3.1 ([12]
). Let Apuq be holomorphic on U pu C q. For any z P R, the fundamental matrices Y r pz, uq, r P Z, of a differential system (1.1) defined in D 1 Ă Dpu 0 q (not necessarily isomonodromic) can be analytically continued w.r.t. u on the whole r τ -cell containing u 0 , maintaining the asymptotics (1.19). The asymptotics is uniform in any compact subset K of the cell, for z Ñ 8 in the sector Moreover, the continuation -maintaining for each u the asymptotics in S r puq -can be extended along any curve slightly beyond the boundary of the cell, if the curves crosses the boundary at a point corresponding to just one Stokes ray crossing arg z " r τ (simple crossing) A second issue must be considered. Suppose that u moves out of the cell along a curve, which then crosses W pr τ q. We face two problems.
-One problem arises when the crossing occurs at a point of ∆. In this case, the coefficients F k puq in Y F pz, uq have poles at those u i´uj " 0, because The actual solutions Y r pz, uq -if they can be extended analytically outside the cell -are in general multivalued for loops around u i´uj " 0 and diverge at u i´uj " 0. The reader can find some concrete example, explicitly worked out, in the proceedings of a Conference in Pisa, 2017 [14].
-Another problem regards the asymptotics. Indeed, the asymptotic behaviour of Y r pz, uq in prescribed sectors S r pD 1 q, S r pDpu 0 qq or S r pKq (all have angular amplitude greater than π) does no longer hold when u is outside the cell containing u 0 .
The issue above can be solved in the strong isomonodromic case when the vanishing conditions (3.1) hold. Notice that by Proposition 3.1 it makes sense to talk about the solutions Y r pz, uq in the whole r τ -cell of u 0 , so that we can define strong isomonodromic deformations in domains compactly contained in the r τ -cell. Accordingly, let the deformation be strongly isomonodromic in Dpu 0 q compactly contained in the r τ -cell. Then, by Definition 2.1, dH r " 0 and dH " 0. Equivalently, by Proposition 2.2, for one value of r (and so for all) dS r " dS r`1 " 0, dB " 0, and moreover there is one Y p0q pz, uq in Levelt form such that dL " 0, dC r " 0. By Proposition 2.1, strong isomonodromicity is equivalent to the fact that the Y r and Y p0q satisfy dY r " ωY r , dY p0q " ωY p0q , (3.4) with form (2.1). Thus, since (3.4) is linear, and its coefficients are holomorphic on u P U pu C q if and only if conditions (3.1) hold, the following proposition follows. As far as the formal fundamental matrix Y F pz, uq is concerned, we have the following Proposition 3.3. Let Apuq be holomorphic on U pu C q and assume that the vanishing conditions (3.1) hold in U pu C q. Then, the coefficients F k puq, k ě 1, of the formal solution Y F pz, uq are holomorphic on U pu C q.
Proof: The coefficients F k puq are computed recursively from (1.1). This standard computation [33] yields coefficients depending rationally on Apuq and on the differences pu i´uj q, which appear in the denominators (see (3.2)). In particular (1.22) holds: Thus, if (3.1) holds, F 1 puq is holomorphic in U pu C q. By Proposition 3.1, the asymptotic expansion is uniform in compact subsets of a cell, so we can substitute it into B i Y " ω i Y , compare coefficients of z´l and find rF l`1 puq, E i s " rF 1 puq, E i sF l puq´B i F l puq, l ě 1, (3.5) with, Moreover, diagpF l`1 q is determined by l pF l`1 q ii puq "´ÿ j‰i A ij puqpF l q ji puq.
(3.6) Therefore, (3.5)-(3.6) recursively determines F l`1 as a function of F l , F l´1 , ..., F 1 . Since F 1 is holomorphic when conditions (3.1) hold, by induction all the F l`1 puq are holomorphic. l Remark 3.1. Proposition 3.3 holds only in the isomonodromic case. In the non-isomonodromic case the vanishing conditions (3.1) only guarantee that F 1 puq is holomorphic. In order for F 1 , F 2 , ..., F l to be holomorphic up to a certain l, also the following quantitieś It remains to check what happens to the asymptotic behaviour of the matrices Y r pz, uq. This requires a certain amount of non-trivial work, which we have done in [12] for a system of the form without assuming that the system is isomonodromic. The series above is assumed to converge at z " 8 with holomorphic matrix coefficients A j puq on U pu C q. The asymptotic theory at z " 8 for (3.7), with u P D 1 sufficiently small contained in a r τ -cell, is the same as for (1.1). Namely, there is a unique formal solution Y F pz, uq " F pz, uqz Bpuq e zΛ , Bpuq " diagpApuqq F pz, uq " I`8 ÿ k"1 F k puqz´k, and unique actual solutions Y r pz, uq.
In order to proceed, we need to take sufficiently small, as follows. Consider the sub-class of Stokes rays associated with the pairs of eigenvalues u i and u j , with label i, j corresponding to components of u C satisfying u C i ‰ u C j . If is small enough 5 , these rays do not cross any admissible direction r τ`hπ when u varies in U pu C q. We define a sector p S r puq Ă R, which contains the "half-plane" r τ´pr´2qπ ă arg z ă r τ`pr´1qπ and extends up to the nearest Stokes rays lying outside the "half-plane" and taken in the sub-class above. Then, we define p S r :" The sectors p S r have angular opening greater than π. The following theorem has been proved in [12]. It requires a non-trivial amount of work, which we necessarily skip here.

Theorem 3.1 ([12]
). Consider the differential system (3.7), with coefficients Apuq, A j puq holomorphic on U pu C q, where is specified as above. Assume that all the F k puq are holomorphic on U pu C q. Moreover, assume that the fundamental matrices which are holomorphic on a r τ -cell by Proposition 3.1, admit analytic continuation on the whole U pu C q as single valued holomorphic functions of u, for any r P Z and z fixed. Then the following results hold. 5 It suffices to take less than the minimum over i and j, such that u C i ‰ u C j , of the distances between the two parallel lines in the complex plane, one passing through u C i and one through u C j , with direction 3π{2´r τ (mod π, or mor 2π, which is the same).
‚ The asymptotic representation of Y r pz, uq, given by the formal matrix Y F pz, uq " F pz, uqz Bepuq e zΛ , extends beyond the r τ -cell, namely uniformly in every compact subset of U 1 pu C q for every 1 ă . Moreover, the Stokes matrices S r puq satisfy for any r P Z pS r q ij " pS r q ji " 0 for i, j such that u C i " u C j .
‚ The system (3.7) at fixed u " u C , namely ‚ In particular, there exists a formal solutionY F pzq of (3.8) satisfying The corresponding unique actual solutionsY r pzq satisfẙ LetS r be the Stokes matrices of the above solutionsY r pzq. Then, lim uÑuc S r puq "S r .
Theorem 3.1 is at the core of the validity of one of the main results of [12], namely Theorem 3.2 below. Indeed, by Propositions 3.2 and 3.3, the assumptions of Theorem 3.1 hold in the strong isomonodromic case if the vanishing conditions (3.1) hold. This yields the following Theorem 3.2 ([12]). Let Apuq be holomorphic on U pu C q and small as specified above. If the system is strongly isomonodromic on a disk interior to a r τ -cell and the vanishing conditions (3.1) hold, then: ‚ the coefficients of the unique formal solution Y F pz, uq " F pz, uqz B e zΛ are holomorphic on U pu C q, and the corresponding actual solutions Y r pz, uq extend holomorphically on RˆU pu C q, maintaining the asimptotics uniformly in every compact subset of U 1 pu C q (@ 1 ă ). 6 Without the assumptions of the theorem, the formal solutions have a more complicated structure. See [7], [8] for a general theory, and [12] for the specific case here studied.
‚ Moreover, for any r, the essential monodromy data S r , S r`1 , B, L, D and C r are constant on the whole U pu C q and satisfy pS r q ij " pS r`1 q ij " pS r q ji " pS r`1 q ji " 0 for i, j s.t. u C i " u C j . ‚ They coincide with a set of essential monodromy data of This means that we take the formal solution of (3.10) satisfying and the associated actual solutionsY r pzq, with Stokes matricesS r . Then, we choose a fundamental matrix solution of (3.10) in Levelt formY p0q pzq "p Y pzqz D zL, and the connection matrixC r such thatY r pzq "Y p0q pzqC r . Then, there exists a fundamental matrix Y p0q pz, uq of (3.9) in Levelt form such that the essential monodromy data S r , S r`1 , B, L, D, C r of (3.9) on the whole U pu C q are preciselẙ S r ,S r`1 , B,L, D,C r .

‚ If the diagonal entries of Apu c q do not differ by non-zero integers, then there is a unique formal solution
The final statement of the theorem says that we just need to compute the fundamental matricesY r pzq, Y r`1 pzq of (3.10), which are asymptotic toY F pzq " Y F pz, u C q in p S r pu C q, and we need to compute a solution Y p0q pzq "p Y pzqz D zL of (3.10) in Levelt form. Then, their essential dataS r ,S r`1 , B,L, D andC r are exactly the desired data S r , S r`1 , B, L, D and C r on the whole U pu C q. There is clearly a problem, namely that we cannot say which is the formal solution satisfyingY F pzq " Y F pz, u C q without knowing already Y F pz, uq. This problem does not exist when the diagonal entries of Apu C q do not differ by non-zero integers, because in this case (3.10) only has the unique formal solution Notice that the Stokes matrices are completely determined by (3.10) in this case.
Two remarks are in order here. First, as already notices, the choice of a solution in Levelt form is not unique, so that there is a freedom in the data C r , D, L. Only the Stokes matrices are uniquely determined. Second, the final statement of Theorem 3.2 is justified by proving that that for any fundamental matrix Y p0q pzq "p Y pzqz D zL of (3.10) in Levelt form, there exists Y p0q pz, uq " p Y pz, uqz D zL satisfying dY " ωpz, uqY (namely an isomonodromic fundamental matrix of (3.9)) and such that Y p0q pz, u C q "Y p0q pzq. We have proved this fact in [12].
The above theorem is useful both for computations and theoretical purposes. As far as computations are concerned, suppose that p Apuq is given in a neighbourhood of a coalescence point u C and that we want to compute monodromy data. This computation is highly transcendental and in general cannot be done at a generic u. But it may happen that it can be explicitly done at the coalescence point, because the system simplifies thanks to the vanishing conditions By Theorem 3.2, the result so obtained yields the constant monodromy data in a neighbourhood of u C . An example of this has been given in [12] concerning monodromy data of a Painlvevé equation, and in [13] for the monodromy data of the Frobenius manifold associated with the reflection group A 3 . From the theoretical point of view, the theorem allows to compute monodromy data in a neighbourhood of a coalescence point in cases when we miss some information about the system away form u C . Indeed, suppose that the system (3.9) is known only at a coalescence point u C ; namely, we only know the explicit form of Apu C q. Moreover, suppose that for some theoretical reason it is known that in a neighbourhood of u C the unknown Apuq must satisfy the vanishing conditions (3.1). Then, Theorem 3.2 yields the essential monodromy data in a whole neighbourhood of u C without knowing explicitly Apuq for u ‰ u C . This approach has been used in [13] to compute the monodromy data of the quantum cohomology of the Grassmannian Grp2, 4q, and to prove in a completely explicit way (i.e. by computations of the numerical values of the data) a conjecture [17], [20], [13], [15] relating them to exceptional collections in derived categories of coherent sheaves on Grp2, 4q. Notice that the quantum cohomology of almost all Grassmannians is characterized by a coalescence phenomenon [11] and that Theorem 3.2 applies, due to the semisimplicity of these quantum cohomologies. This fact guarantees that the computation of the monodromy data for Grassmannians, which is performed starting from a coealescence point, is justified.

Remark 3.2. For a system analogous to (1.1), associated with a semisimple Frobenius manifold, where
Apuq is skew symmetric, a synthetic proof is given in [20] that a fundamental matrix solution asymptotic to the formal solution in a sector of central opening angle π`ε (the analogous to our Y r pz, uq " Y F pz, uq) is holomorphic in a small neighbourhood of a coalescence point u C . This result, in case of Frobenius manifolds, is analogous of the first point of Theorem 3.2. The proof in [20] is based on the Laplace transformation of the irregular system into an isomonodromic Fuchsian system, whose associated Pfaffian system has known analyticity properties [34].

Appendix I: Weak and strong isomonodromic deformations of Fuchsian systems
The difference between weak and strong isomonodromic deformations naturally arises in case of Fuchsian systems. This is synonymous of Schlesinger and non-Schlesinger deformations studied by Bolibruch. Up to a Möbius transformation, we can assume that z " 8 is non singular. Accordingly, we consider the u-family of nˆn Fuschsian systems depending holomorphically on the parameter u " pu 1 , ..., u N q in a small polydisc Dpu 0 q with center u 0 " pu 0 1 , ..., u 0 N q, contained in C N z Ť i‰j tpu i´uj q " 0u. We take pz, uq in Notice that Dpu 0 q " D 1ˆ¨¨¨ˆDN , where D i is a disk centered at u 0 i and D i X D j " H. Thus, the fundamental group of P 1 ztu 1 , ..., u N u can be defined in a u-independent way. 7 Definition 4.1. The family is weakly isomonodromic if for every u P Dpu 0 q there exists a fundamental matrix Y pz, uq such that it has the same monodromy matrices for all u P Dpu 0 q. The matrix Y pz, uq is then a weak isomonodromic family of fundamental matrices.
It follows from the theorem on analytic dependence on parameters that there exists a family of fundamental matrices r Y pz, uq which is analytic in pz, uq on the universal covering of E, but which is not necessarily isomonodromic. Namely, r Y pz, uq " Y pz, uqCpuq for some connection matrix Cpuq. If M 1 , ..., M N are the 7 In other words, E " P 1ˆD pu 0 qz Ť N i"1 tpz´u i q " 0u can be retracted to P 1 ztu 0 1 , ..., u 0 N u, and π 1 pE; pz 0 , u 0 qq is isomorphic to π 1 pP 1 ztu 0 1 , ..., u 0 N u; z 0 q. monodromy matrices of Y pz, uq w.r.t. a basis of π 1 pP 1 ztu 0 1 , ..., u 0 N u; z 0 q, then r Y pz, uq has non-constant matrices M j puq " Cpuq´1M j Cpuq. For a Fuchsian isomonodromic system, it can be proved that there exists an isomonodromic Y pz, uq such that Cpuq is holomorphic. Namely: [5]). If (4.1) is weakly isomonodromic, then there exists an isomonodromic family of fundamental matrices Y pz, uq which is holomorphic in z and u on the universal covering of E.
The proof makes use of holomorphic bundles, and we refer to [4], [5].
Proof: Condition 2) is Frobenius integrability condition for the Pfaffian system dY " ωY. Condition 1) assures that the z part of the Pfaffian system is the Fuchsian system (4.1). If the deformation is isomonodromic, than by Proposition 4.1 there is a holomorphic family Y pz, uq and as in the case of (1.5)-(1.6), we see that ω " dY¨Y´1 is single valued and holomorphic on E. This also implies that the Pfaffian system is integrable, then 2) holds. Conversely, if 2) holds, then by the general properties of linear Pfaffian systems there exist a solution Y , holomorphic on the universal covering of E, whose monodromy -as in (1.5)-(1.6) -is independent of u. l We are going to show below that for a non-resonant Fuchsian system (i.e. the eigenvalues of the matrices A i do not differ by non-zero integers) all weak isomonodromic deformations are actually strong, namely not only the monodromy matrices are constant, but also certain essential monodromy data do not depend on u (see Definition 4.2 below). On the other hand, if the system is resonant, there exist both weak and strong isomonodromic deformations, which are inequivalent. To say it in other well known words, in the non-resonant case only Schlesinger deformations exist, while in the resonant case also non-Schlesinger deformations appear. The former are strong (preserving essential data), the latter are weak (preserving only monodromy matrices). First, let us recall Schlesinger deformations.

Schlesinger deformations: For a Fuchsian system
it is always possible to consider its Schlesinger deformations, given by ω " ω s , where ω s :"

Such deformations exist if and only if there exist
A standard computation shows that the condition dω s " ω s^ωs is equivalent to the Schlesinger equations This implies that which ensures that ř i A i puq " 0 whenever ř i A i pu 0 q " 0. The Schlesinger system is well known to be Frobenius integrable. We conclude that ω s , called Schlesinger deformation of the Fuchsian system (4.2), always exists, including the resonant case (but it is not the unique one in this case). For an isomonodromic holomorphic fundamental matrix Y s pz, uq of ω s , we have so that Y s p8, uq is a constant independent of u (for example Y s p8, uq " I). For this reason, we say that the Schlesinger deformation ω s is normalized. l Non-normalized Schlesinger deformations [24]: Together with ω s , there always exist deformations with holomorphic matrix coefficients γ i puq on Dpu 0 q, which are called non-normalized. To see this, let Y s pz, uq be such that Y s p8, zq " I (or another constant matrix) and dY s " ω s Y s . Then let p Y pz, uq :" ΓpuqY s pz, uq, for a holomorphically invertible matrix Γpuq on Dpu 0 q (so that p Y p8, uq " Γpuq and the normalization at 8 is lost). We have Conversely, for any γ 1 , ..., γ N we need to prove that there exists an holomorphically invertible Γ such that dΓ¨Γ´1 " The integrability condition of this system is On the other hand, by assumption, ω " ω s`ř i γ i da i satisfies dω " ω^ω. We have and ω^ω " ω s^ωs`ÿ iăj pγ i γ j´γj γ i qda i^d a j .
Using that dω s " ω s^ωs , we see that dω " ω^ω implies (4.3). l The same considerations as above show that if ω is a 1-form for an isomonodromic family of Fuchsian systems, then any ω`ř i γ i puqda i is again responsible for an isomonodromic deformation.
Schlesinger deformations have the following property.

4)
where the matrices A i puq and γ i puq are holomorphic in Dpu 0 q. Then each A i puq is holomorphically similar to a constant Jordan form J i . Namely, there exists G i puq holomorphic in Dpu 0 q such that J i " G i puq´1A i puqG i puq is Jordan and independent of u. In particular, the spectrum of each A i puq is independent of u, so that a Schlesinger isomonodromic deformation is isospectral.
The proof is like the proof of Proposition 1.1. Indeed, suppose we want to prove that J 1 " G 1 puq´1A 1 puqG 1 puq. As far as the local behaviour around z´u 1 " 0 is concerned, (4.4) can be rewritten as where the N`1 independent variables are x :" z´u 1 ; u 1 , u 2 , ..., u N and each B k px, uq, for k " 0, 1, ..., N , is holomorphic in a neighbourhood of x " z´u 1 " 0. So, the proof repeats the arguments of the proofs of Lemma 1.1 and Proposition 1.1 (see again [23], [34], [3]) We now report the most general ω for Theorem 4.1. Let Y pz, uq be a holomorphic weak isomonodromic fundamental matrix solution of dY " ωY , where ω is as in Theorem 4.1. Being a solution of a Fuchsian system, we write its Levelt form at where C i puq is an invertible connection matrix, and at each value of u we have that   In other words, the diagonal of L i puq is constant and D i is constant. Thus, an isomonodromic deformation of a Fuchsian system is "isospectral".
We observe that if C i and L i are constant, then ω " ω s . Indeed, we have The holomorphic part is regular at z " u i , and moreover dY Y´1 " 0 at z " 8, having fixed Y p8, uq " I. Thus, Liouville Theorem yields ω " ω s . The converse of the above is also true: , [34], [5] and [26]). If the deformation is Schlesinger, i.e. ω " ω s (or its non-normalized version), then dC i " 0 and dL i " 0 are constant. Moreover, we leave it as an exercise to show that dC j " 0 implies dL j " 0 (it suffices to notice that we can take the monodromy matrices M j " C´1 j e 2πiLj C j ). Thus, one can define a strong deformation to be one such that all the C j are constant, in the same way as in Definition 2.1 (the connection matrices C j play the same role of the H r and H).
Thus, we conclude that

Proposition 4.4 (Proposition 4.3 revised). An isomonodromic deformation is strong if and only if it is a Schlesinger deformation (or its non-normalized version).
Following [26], we may call P i pz, uq :" pz´u i q Di pz´u i q Lipuq C i puq the principal part of Y pz, uq at z " u i , so that Y pz, uq " p Y i pz, uqP i pz, uq. In case C i and L i are constant, then P i pz, uq " P i pz´u i q, namely, P i depends on pz, uq only through the combination pz´u i q. If such a dependence occurs at any u i , the deformation is called isoprincipal in [26]. The main result of [26] is that a family of Fuchsian systems (4.1) is an isoprincipal deformation if and only if the matrices A i satisfy the Schlesinger equations. This result is equivalent to Proposition 4.3.
We are ready to review the most general form of ω.
Theorem 4.2 (Non-Schesinger deformations -Bolibruch [4], [5]). All the possible matrix differential 1form of Theorem 4.1, holomorphic on P 1ˆD pu 0 qz Ť n i"1 tz´u i " 0u, which give an isomonodromic family of Fuchsian systems (4.1), have the form where γ i puq, γ ijk puq are holomorphic on Dpu 0 q and m i is the maximal integer difference of eigenvalues of A i puq.
Proof: We take the Schlesinger deformation ω s with the same initial condition Let Y s pz, uq be an isomonodromic fundamental matrix for ω s . By isomonodromicity, Y pz, uq and Y s pz, uq have the same monodromy matrices, which are those of Hence, Γpz, uq :" Y pz, uq Y s pz, uq´1 is single valued and meromorphic on P 1ˆD pu 0 q, with poles at z " u i , 1 ď i ď N . Moreover ω " dΓ¨Γ´1`Γω s Γ´1. This expression does show that ω has the structure (4.6). In order to predict m i , we write where L s i and C s i are constant, by Proposition 4.3. Being the monodromy M i of Y and Y s the same, we have Thus, and R i must have the same block structure of L i puq and L s i . Moreover, Thus, we have The last term p Y i pz´u i q Di dR i¨R´1 i pz´u i q´D i p Y´1 i contains matrix entries with with poles at pz´u i q " 0, plus holomorphic terms. This proves again, by Liouville theorem, that ω has the structure (4.6). The block diagonal structure of R i , being the same as that of D i " D piq 1 '¨¨¨' D piq i , assures that m i is the maximum over q of the maximal difference of the eigenvalues of D piq q , q " 1, 2, ..., i . This is precisely the maximal integer difference of eigenvalues of A i puq. l In the non-resonant case, m i " 0 for every i " 1, ..., N . Therefore, we obtain the following There are examples of non-Schlesinger deformations in [5], [27] and [2]. 8 A simple example is given at the end of [26], and we report it here. Consider the family of differential systems Here hpuq is an arbitrary function, which is holomorphic at u " 0. The system is resonant at all the Fuchsian singularities z " u, 1, 2, 3. One can check that the residue matrices do not satisfy the Schlesinger equations, but the u-deformation is isomonodromic in the weak sense, because Y pz, uq is single-valued and hence has trivial monodromy matrices M i " I :"diagp1, 1q. For z close to u, we have So, the deformation is not isomonodromic in the strong -or isoprincipal -sense. The monodromy matrix is nevertheless constant and equal to I. As noted in [26], the example shows that a weak isomonodromic deformation may not have the Painlevé property. Indeed, in the above example we can choose hpuq arbitrarily (provided it is holomorphic at u " 0). On the other hand, if the deformation is isomonodromic in the strong sense, namely Schlesinger, then the Painlevé property holds also in the resonant case.

Appendix II: Proof of Proposition 1.2
We recall that a small polydisc D 1 has been introduced by Sibuya to guarantee that the fundamental matrices Y r pz, uq in (1.18) are holomorphic of u. Lemma 5.1. Let ω 0 pxq "`Λ`A z˘b e as in (1.3), let ω 1 , ..., ω n be holomorphic of pz, uq P CˆDpu 0 q, and Apuq holomorphic on Dpu 0 q. Assume that z " 8 is at most a pole of ω 1 , ..., ω n . Then, for u P D 1 , the Pfaffian system has the following structure ω "ˆΛ`A z˙d z`n ÿ j"1 pzE j`ωj p0, uqqdu j , with ω j p0, uq " rF 1 puq, E j s`D j puq, (5.1) where F 1 , appearing in the formal expansion (1.17), is given explicitly in (1.22), and D j puq " BH r puq Bu j H r puq´1 8 One can also have a look at [28], though the question there is concerned with Schlesinger transformations that also shift the eigenvalues.
is a diagonal matrix independent of r P Z and holomorphic on u P D 1 . Moreover is polynomial in z of order Opzq, with coefficients depending holomorphically on u P Dpu 0 q. Finally, we have dB " 0.
in the sector S r pD 1 q. The asymptotics holds for every r P Z. Now, the asymptotic expansion of a function (in our case ω j ) on a sector is unique. Since S r pD 1 q X S r`1 pD 1 q is not empty, this implies that ω j has the same expansion in every sector S r pD 1 q, @r. In particular, B j H r H´1 r cannot depend on r. We set D j puq :" BH r puq Bu j H´1 r puq @r P Z.
Moreover, being ω j single valued, the asymptotic series represented by O`1 z˘a bove, must be a convergent Taylor expansion. In conclusion ω j´p zE j`r F 1 , E j s`D j q " Oˆ1 z˙i s holomorphic at z " 8 Keeping into account (5.2) and Liouville Theorem, we obtain that ω j´p zE j`r F 1 , E j s`D j q " 0. 2) A satisfies the non-linear system BA Bu j " rω j p0, uq, As, j " 1, 2, ..., n, (5.4) and the system (5.4) is Frobenius integrable.
The above (5.3) determines ω j p0, uq up to a diagonal matrix D j puq, as follows ω j p0, uq "ˆA ab puqpδ aj´δbj q u a´ub˙n a,b"1`D j puq. (5.5) Moreover, diagpA 11 , ..., A nn q is constant and where D is a matrix whose diagonal only depends on u P Dpu 0 q.