Isomonodromic Laplace Transform with Coalescing Eigenvalues and Confluence of Fuchsian Singularities

We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters u=(u_1,...,u_n), which are eigenvalues of the leading matrix at the irregular singuilarity. At the same time, we consider a Pfaffian system of non-normalized Schlesinger type expressing isomonodromy of a Fuchsian system, whose poles are the deformation parameters u_1,...,u_n. The parameters vary in a polydisc containing a coalescence locus for the eigenvalues of the leading matrix of the irregular system, corresponding to confluence of the Fuchsian singularities. We construct isomonodromic selected and singular vector solutions of the Fuchsian Pfaffian system together with their isomonodromic connection coefficients, so extending a result of references [4] and [20] to the isomonodromic case, including confluence of singularities. Then, we introduce an isomonodromic Laplace transform of the selected and singular vector solutions, allowing to obtain isomonodromic fundamental solutions for the irregular system, and their Stokes matrices expressed in terms of connection coefficients. These facts, in addition to extending [4] and [20] to the isomonodromic case with coalescences/confluences, allow to prove by means of Laplace transform the main result of reference [11], which is the analytic theory of non-generic isomonodromic deformations of the irregular system with coalescing eigenvalues.


Introduction
In this paper, I answer a question asked when I presented the results of [13] and the related paper [25]. Paper [13] deals with the extension of the theory of isomonodromic deformations of the differential system (1.1), in the presence of a coalescence phenomenon involving the eigenvalues of the leading matrix . These eigenvalues are the deformation parameters. The question is if we can obtain some results of [13] in terms of the Laplace transform relating system (1.1) to a Fuchsian one, such as system (1.4). The latter has simple poles at the eigenvalues of , so that the coalescence of the eigenvalues will correspond to the confluence of the Fuchsian singularities. So the question is if combining integrable deformations of Fuchsian systems, confluence of singularities and Laplace transform, we can obtain the results of [13]. The positive answer is Theorem 7.1 of this paper. In order to achieve it, we extend to the case depending on deformation parameters, including their coalescence, one main result of [4,23] concerning the existence of selected and singular vector solutions of a Pfaffian Fuchsian system associated with (1.4) (see the system (5.3)), and their connection coefficients, which will be isomonodromic. This will be obtained in Theorem 5.1 and Proposition 5.1.
In [13], the isomonodromy deformation theory of an n-dimensional differential system with Fuchsian singularity at z = 0 and singularity of the second kind at z = ∞ of Poincaré rank 1 dY dz = (u) + A(u) z Y , (u) = diag(u 1 , . . . , u n ), (1.1) has been considered 1 . The deformation parameters u = (u 1 , . . . , u n ) vary in a polydisc where the matrix A(u) is holomorphic. One of the main results of [13] is the extension of the theory of isomonodromic deformations of (1.1) to the non-generic case when has coalescing eigenvalues but remains diagonalizable. This means that the polydisc contains a locus of coalescence points such that u i = u j for some 1 ≤ i = j ≤ n. In this case, z = ∞ is sometimes called resonant irregular singularity. On a sufficiently small domain in the polydisc, the well-known theory of isomonodromy deformations applies and allows to define constant monodromy data. Theorem 1.1 and corollary 1.1 of [13] say that these data are well defined and constant on the whole polydisc, including the coalescence locus, if the entries of A(u) satisfy the vanishing conditions (A(u)) i j → 0 when u tends to a coalescence point such that u i −u j → 0 at this point. For future use, we denote by λ 1 , . . . , λ n the diagonal entries of A(u), and B := diag(A(u)) = diag(λ 1 , . . . , λ n ).
We will see that these λ k are constant in the isomonodromic case. From another perspective, if u is fixed and u i = u j for i = j, namely for a system (1.1) not depending on parameters with pairwise distinct eigenvalues of , it is well known that columns of fundamental matrix solutions with prescribed asymptotics in Stokes sectors at z = ∞ can be obtained by Laplace-type integrals of certain selected column-vector solutions of an n-dimensional Fuchsian system of the type Here, E k is the elementary matrix whose entries are zero, except for (E k ) kk = 1. These facts are studied in the seminal paper [4] in the generic case of non-integer diagonal entries λ k of A. The results of [4] have been extended in [23] to the general case, when the entries λ k take any complex value. The purpose of the present paper is to introduce an isomonodromic Laplace transform relating (1.1) to an isomonodromic Fuchsian system (1.4) when u 1 , . . . , u n vary in a polydisc containing a locus of coalescence points. More precisely, the Laplace transform will relate solutions of the integrable Pfaffian systems (2.14) and (5.3) introduced later, associated with (1.1) and (1.4), respectively. The two main goals will be: • Theorem 5.1, which characterizes selected vector solutions and singular vector solutions of (1.4) and (5.3), so extending the results of [4] and [23] to the case depending on isomonodromic deformation parameters, including coalescing Fuchsian singularities u 1 , . . . , u n . • Theorem 7.1, in which the Laplace transform of the vector solutions of Theorem 5.1 allows to obtain the main results (I), (II) and (III) of [13] in the presence of coalescing eigenvalues u 1 , . . . , u n of (u).
In details.
The above is the deformation parameters dependent analogue of the definition of connection coefficients in [23]. • In Proposition 5.1, we prove that the c (ν) jk are isomonodromic connection coefficients, namely independent of u. When there is a coalescence u j = u k in the polydisc, they satisfy c (ν) jk = 0, j = k.
• In Theorem 7.1, the Laplace transform of the vectors k (λ, u |ν) or (sing) k (λ, u |ν) yields the columns of the isomonodromic fundamental matrix solutions Y ν (z, u) of (1.1), labelled by ν ∈ Z, uniquely determined by a prescribed asymptotic behaviour in certain u-independent sectors S ν , of central opening angle greater than π . The analytic properties for the matrices Y ν (z, u) will be proved, so reobtaining the result (I) above. In order to describe the Stokes phenomenon, only three solutions Y ν (z, u), Y ν+μ (z, u) and Y ν+2μ (z, u) suffice. The labelling will be explained later. The Stokes matrices S ν+kμ , k = 0, 1, defined by a relation Y ν+(k+1)μ = Y ν+kμ S ν+kμ , will be expressed in terms of the coefficients c (ν) jk in formula (7.9). This extends to the isomonodromic case, including coalescences, an analogous expression appearing in [4,23] and implies the results in (II) above.
• In Sect. 8, we re-obtain the result (III), that system (1.1), "frozen" by fixing u equal to the most coalescence point u c in the polydisc (see Sect. 2.1 for u c ), admits a unique formal solution if and only if the (constant) diagonal entries λ j of A satisfy λ i − λ j / ∈ Z\{0} for every i = j such that u c i = u c j . In this case we prove that the selected vector solutions of the Fuchsian system (1.4) at u = u c , needed to perform the Laplace transforms, are uniquely determined. On the other hand, if some λ i − λ j ∈ Z\{0} corresponding to u c i = u c j , then there is a family of solutions of the Fuchsian system (1.4) at a coalescence point, depending on a finite number of parameters: this facts is responsible, through the Laplace transform, of the existence of a family of formal solutions at the coalescence point. the selected solutions of (1.4) (which are part of the monodromy of the Dubrovin-Frobenius manifold) are expressed in terms of the entries of the Stokes matrices. See also [21,61].
In proposition 2.5.1 of [22], the authors prove (I) when system (1.1) is associated with a Dubrovin-Frobenius manifold with semisimple coalescence points, and A is skew-symmetric (in [22] the irregular singularity is at z = 0). Their proof contains the core idea that the analytic properties of a solution Y (z, u) in (I) are obtainable, by a Laplace transform, from the analytic properties of a fundamental matrix solution (λ, u) of the Fuchsian Pfaffian system associated with (1.4) (see their Lemma 2.5.3). The latter is a particular case of the Pfaffian systems studied in [63]. On the other hand, the analysis of selected and singular vector solutions of the Fuchsian Pfaffian system, required in our paper to cover all possible cases (all possible A), is not necessary in [22], due to the skew-symmetry of A, and the specific form of their Pfaffian system (see their equation (2.5.2); their discussion is equivalent our case λ j = −1 for all j = 1, . . . , n). Moreover, points (II) and (III) are not discussed in [22] by means of the Laplace transform.
In the present paper, by an isomonodromic Laplace transform, we prove (I), (II) and (III), and at the same time we generalize the results of [4,23] to the isomonodromic case with coalescences, with no assumptions on the eigenvalues and the diagonal entries of A. This analytic construction, to the best of our knowledge, cannot be found in the literature.
The approach of the present paper may also be used to extend the results of [19,20] described above, relating the deformed flat connection and the intersection form, namely Stokes matrices and monodromy group of the Dubrovin-Frobenius manifold, in case of semisimple coalescent Frobenius structures studied in [10,14,15,17].
Stickily related to ours are the important results of [52]. In [13] (and in the present paper by Laplace transform), we have answered the question if the integrable deformation (2.14) of system (1.1) extends from a polydisc (or a small open set) not containing coalescence points to a wider domain intersecting (a stratum of) the coalescence locus, and we have characterized the monodromy data. The converse question is answered in [52], namely if an integrable deformation (2.14) of ( (u c ) + A(u c )/z)dz exists and is unique, having formal normal form d(z (u)) + B/z dz, where B is the diagonal of A(u c ). More broadly, the question of [52] is the existence and uniqueness of integrable deformations of meromorphic connections on P 1 with irregular singularity, when a prefixed restriction is given at a single point t o in the space of deformation parameters T , allowed to be a degenerate point, namely a coalescence point in our case (in [52], deformation parameters are called t ∈ T ). One asks if a connection ω(z, t 0 ) given at t o ∈ T can be deformed to ω(z, t), and if this deformation is unique. 2 Concerning uniqueness, for a fixed normal form ω 0 (z, t), the problem is to classify isomorphism pairs (ω, G) consisting of an integrable connection ω(z, t) (with poles in T × {z = 0}, being z = 0 used in [52], while z = ∞ is used in our works) and a formal gauge transformation G(z, t) (formal in z but holomorphic in t), transforming ω(z, t) to ω 0 (z, t). In a general context, a uniqueness theorem is proved in [60]: two pairs are isomorphic (meaning that the composition of a gauge of one pair with the inverse gauge of the other pair is convergent w.r.t. z) if and only if their restriction to any specific value t o are isomorphic. Thus, the t-extension of a pair in a neighbourhood of t 0 is unique up to isomorphism. The proof in [60] makes use of the results of Kedlaya [34,35] and Mochizuki [45][46][47][48], which allows to blow up T × {0}, and of the higher-dimensional asymptotic analysis in poli-sectors for the formal gauge transformations, that is Majima's asymptotic analysis [40] for Pfaffian systems with irregular singularities. In [52], the uniqueness result is proved for a restricted class of integrable connections, in which our (2.14) is contained (with irregular singularity at z = 0 instead of ∞). So, given a block-diagonal normal form ω 0 (z, t) and a pair consisting of ω(t o , z) and a formal gauge G(t o , z), it is proved that the pair can be deformed (existence) in a unique way (uniqueness) to ω(z, t), G(z, t), such that G[ω] = ω 0 . The strategy is to use a sequence of Kedlaya-Mochizuki blow-ups to obtain a good normal form (see also [50,51]). Then, Majima results on asymptotic analysis can be used and adapted. In our specific case, theorem 4.9 of [52] means the existence and uniqueness of the integrable deformation (2.14) of ( (u c ) + A(u c )/z)dz, formally equivalent to d(z (u)) + B/zdz. These facts generalize results of Malgrange [41,42] for irregular singularities to the case of coalescence points. Theorem 4.9, obtained in [52] in geometric terms, has been successively proved in [11] by analytic methods. In [11], the integrable deformation is obtained from prefixed monodromy data at a coalescence point, using the analytic L p theory a Riemann-Hilber boundary value problems. Both authors of [52] and [11] apply their results to semisimple Dubrovin-Frobenius manifolds. In particular, [11] proves that any formal semisimple Frobenius manifold is the completion of a pointed germ of an analytic Dubrovin-Frobenius manifold. The result is extended to F-manifolds in the recent work [12].
A geometric formulation of the Laplace transform we have used here, together with a synthetic proof of part of Theorem 1.1 of [13], is the object of the recent work [53].

Review of background material
This section contains known material to motivate and understand our paper. For X a topological space, we denote by R(X ) its universal covering. For α < β ∈ R, a sector is written as S(α, β) := {z ∈ R(C\{0}) such that α < arg z < β}.

Background 1: isomonodromy deformations of (1.1) with coalescing eigenvalues
We review some results of [13,25] (see also [16,24,26]). Consider a differential system (1.1) with an n × n with matrix coefficient A(u) holomorphic in a polydisc The eigenvalues of (u) coalesce at u c and also along the following coalescence locus We assume that D(u c ) is sufficiently small so that u c is the most coalescent point. Namely, if u c j = u c k for some j = k, then u j = u k for all u ∈ D(u c ). A more precise characterization of the radius 0 of the polydisc will be given in Sect. 5. For be a (smaller) polydisc centered at u 0 , not containing coalescence points.

Deformations in D(u 0 )
If D(u 0 ) is sufficiently small, the isomonodromic theory of Jimbo, Miwa and Ueno [33] assures that the essential monodromy data of (1.1) (see Definition 2.1) are constant over D(u 0 ) and can be computed fixing u = u 0 .

Fig. 1
Successive sectors S ν (D(u 0 )) and S ν+μ (D(u 0 )). Their intersection (in the right part of the figure) does not contain Stokes rays. It contains the admissible direction arg z = τ (0) that we decide to label from 0 to μ (0) − 1. They are basic rays, since they generate all the Stokes rays in R(C\{0}) associated with (u 0 ) by the formula The choice to label a specific Stokes ray with 0, as τ 0 above, is arbitrary, and it induces the labelling ν ∈ Z for all other rays. Suppose the labelling has been chosen. Then, for some ν ∈ Z, we have Equivalently, given τ (0) , one can choose a ν and decide to call τ ν and τ ν+1 the Stokes rays satisfying (2.3). This induces the labelling of all other rays (notice that μ (0) is not a choice!). Similarly, we consider the Stokes rays ((u j − u k )z) = 0, ((u j − u k )z) < 0 of (u). If D(u 0 ) is sufficiently small, when u varies the Stokes rays of (u) rotate without crossing arg z = τ (0) mod π . For k ∈ Z, we take the sector S τ (0) + (k − 1)π, τ (0) + kπ and extend it in angular amplitude up to the nearest Stokes rays of (u) outside. The resulting (open) sector will be denoted by S ν+kμ (0) (u), and we define The reason for the labelling is that S By construction, S ν (D(u 0 )) has central angular opening greater than π . See Fig. 1. [30,54,55]; see also [13,25,33]). Let D(u 0 ), not containing coalescence points, be sufficiently small so that the Stokes rays of (u) do not cross 3 the admissible rays arg z = τ (0) + hπ , h ∈ Z, as u varies in D(u 0 ). System (1.1) has a unique formal solution is a formal series, with holomorphic matrix coefficients F k (u).For every ν ∈ Z, there exist unique fundamental matrix solutions , such that uniformly in u ∈ D(u 0 ) the following asymptotic behaviour holds The coefficients F k are computed recursively [13,62] Holomorphic Stokes matrices S ν (u), ν ∈ Z, are the connection matrices defined by Notice that S ν (D(u 0 )) ∩ S ν+μ (0) (D(u 0 )) does not contain Stokes rays of (u), for every u ∈ D(u 0 ). At every fixed u ∈ D(u 0 ), system (1.1) admits a fundamental matrix solution in Levelt form where the series is convergent absolutely in every ball |z| < N , for every N > 0.
Here, D is diagonal with integer entries (called valuations), L has eigenvalues with real part lying in [0, 1), and D + lim z→0 z D Lz −D is a Jordan form of A. A central A pair of Stokes matrices S ν , S ν+μ (0) , together with B, C ν and L are sufficient to calculate all the other S ν and C ν , for all ν ∈ Z (see [1,13]). The monodromy matrices at z = 0 are for Y (0) and Y ν , respectively. Hence, it makes sense to define strong isomonodromy deformations, as follows.
Definition 2.1 Fixed a ν ∈ Z, we call essential monodromy data the matrices The deformation u is strongly isomonodromic on D(u 0 ), if the essential monodromy data are constant on D(u 0 ).
We introduced the terminology strong in [25], to mean that all the essential monodromy data are constant, contrary to the case of weak isomonodromic deformations, which only preserve monodromy matrices of a certain fundamental matrix solution. For a deformation to be weakly isomonodromic it is necessary and sufficient that (1.1) is the z-component of a certain Pfaffian system dY = ω(z, u)Y , Frobenius integrable (i.e. dω = ω ∧ ω). If ω is of very specific form, the deformation becomes strongly isomonodromic, according to the following with the matrix coefficients (here F 1 is in (2.8)) The above theorem is analogous to the characterization of isomonodromic deformations in [33], but includes also possible resonances in A (see [13] and Appendix B of [25]). Notice that ω(z, u) in (2.14)-(2.15) has components . . , n.

Deformations in D(u c ) with coalescences
When the polydisc contains a coalescence locus , the analysis presents problematic issues.
may be singular at , namely the limit for u → u * ∈ along any direction may diverge, and is in general a branching locus [43]. • The monodromy data associated with a fundamental matrix solutionY (z) of differ from those of any fundamental solution Y (z, u) of (1.1) at u / ∈ ( [2,3,13]).
In R(C\{0}), we introduce the Stokes rays of (u c ) and an admissible direction at u c arg z = τ, (2.20) such that none of the Stokes rays at u = u c take this direction. Notice that τ is associated with u c , differently from τ (0) of Sect. 2.1.1. We choose μ basic Stokes rays of (u c ). These are all and the only Stokes rays lying in a sector of amplitude π , whose boundaries are not Stokes rays of (u c ). Notice that μ is different from μ (0) used in Sect. 2.1.1. We label their directions arg(z) as follows: The directions of all the other Stokes rays of (u c ) in R(C\{0}) are consequently labelled by an integer ν ∈ Z arg z = τ ν := τ ν 0 + kπ, with ν 0 ∈ {0, . . . , μ − 1} and ν := ν 0 + kμ. (2.21) They satisfy τ ν < τ ν+1 . Analogously, at any other u ∈ D(u c ), we define Stokes rays ((u i − u j )z) = 0, ((u i − u j )z) < 0 of (u). They behave differently from the case of D(u 0 ). Indeed, if u varies in D(u c ), some Stokes rays cross the admissible directions arg z = τ mod π , as follows. Let i, j, k be such that u c i = u c j = u c k . Then, as u moves away from u c , a Stokes ray of (u c ) characterized by ((u c i − u c k )z) = 0 generates three rays.
is sufficiently small (as in (5.1)), they do not cross arg z = τ mod π as u varies in D(u c ). The third ray is ((u i − u j )z) = 0. When u varies in D(u c ) making a complete loop (u i − u j ) → (u i − u j )e 2πi around the locus {u ∈ D(u c ) | u i − u j = 0} ⊂ , the third ray crosses arg z = τ mod 2π and arg z = τ − π mod 2π . This identifies a crossing locus X (τ ) ⊂ D(u c ) of points u such that there exists a Stokes ray of (u) (so infinitely many in R(C\{0})) with direction τ mod π . Thus, the choice of τ induces a cell decomposition of D(u c ). Each cell is called τ -cell. If u varies in the interior of a τ -cell, no Stokes rays cross the admissible directions arg z = τ + hπ , h ∈ Z, but if u varies in the whole D(u c ), then X (τ ) is crossed, and thus Proposition 2.1 does not hold.
To overcome this difficulty, we first take a point u 0 in a τ -cell, and consider a polydisc D(u 0 ) contained in the τ -cell, satisfying the assumptions of Sect. 2.1.1. Accordingly, we can define as before the sectors S ν+kμ (u) of angular amplitude greater than π , and Notice that here we are using τ and μ in place of τ (0) and μ (0) . With the above sectors, monodromy data in (2.11)-(2.13) can be defined in D(u 0 ).  The extension of the theory of isomonodromy deformations on the whole D(u c ) is given in [13] by the following theorem, which is a detailed exposition of the points (I) and (II) of Introduction, while point (III) is expressed by Corollary 2.1.

Theorem 2.2 ([13]). Let A(u) be holomorphic on
Then, the following statements hold. Part I.
uniformly in u ∈ D(u c ), for z → ∞ in a wide sector S ν containing S ν (D(u 0 )), to be defined later in (7.3).

5) The Stokes matrices satisfy the vanishing conditions
The assumption of Corollary 2.1 will be called partial non-resonance. If it holds, (II,1) says that in order to obtain the essential monodromy data of (1.1) it suffices to computeS ν ,S ν+μ ,L,C ν andD for (2.19), which is simper than (1.1), because This allows in some cases the explicit computation of monodromy data. An important example with algebro-geometric implications can be found in [14].

Remark 2.1
The following statement, not mentioned in [13], holds.
If (1.1) is an isomonodromic family on the polydisc minus the coalescence locus, in the sense that dY = ωY in (2.14)- I thank the referee for suggesting to write the above statement. The sketch of the proof is as follows: integrability dω = ω ∧ ω on D(u c )\ implies (2.16), namely We want to prove that A i j (u) → 0 for u i − u j → 0, for i = j. From (2.23) and (2.18) we explicitly obtain The left-hand side is holomorphic everywhere on D(u c ) by assumption on A, and so must be the right-hand side. This implies that holomorphically Then, Theorem 2.2 holds and we conclude.
The difficulty in proving Theorem 2.2 is the analysis of the Stokes phenomenon at z = ∞. On the other hand, coalescences does not affect the analysis at the Fuchsian singularity z = 0, so it is not an issue for the proof of the statements concerning Y (0) (z, u), L , D and C ν (as far as the contribution of Y (0) is concerned). See Proposition 17.1 of [13], and the proof of Theorem 4.9 in [25]. For this reason, in the present paper we will not deal with Y (0) (z, u), L , D, C ν and (II,3)-(II,4) above.

Background 2: Laplace transform, connection coefficients and Stokes matrices
In this section, we fix u ∈ D(u c )\ . Accordingly, system (1.1) is to be considered as a system not depending on deformation parameters, with leading matrix having pairwise distinct eigenvalues, and system (1.4) is equivalent to (1.3), which does not depend on parameters. For simplicity of notations, let us fix for example u = u 0 , as in Section 2.1.1.
Solutions Y ν (z) of (1.1) with canonical asymptotics Y F (z) (u = u 0 fixed is not indicated) can be expressed in terms of convergent Laplace-type integrals [5,31], where the integrands are solutions of the Fuchsian system 5 Indeed, let (λ) be a vector valued function and define where γ is a suitable path. Then, substituting in (1.1), we have

Multiplying to the left by
(2.26) In order to define matrix solutions of of (2.26) as single valued functions, we consider the λ-plane with branch-cuts. Let η (0) ∈ R satisfy Fig. 2. Conditions (2.27) mean that a cut L k does not contain another pole u 0 j , j = k. For this reason η (0) is called admissible direction at u 0 . Then, we choose a branch of the logarithms ln(λ−u 0 Following [4], the λ-plane with these cuts and choices of the logarithms is denoted by P η (0) . Matrix solutions of (2.26) are well defined as single-valued functions of λ ∈ P η (0) .

Remark 2.2 P η (0)
can be identified with one of the countably many components of Each component is obtained by a deck transformation starting from one. Fix one component, for example P η (0) , and define 2n letters l k := cross a lift of L k from the right, where "right" or "left" refers to the orientation of L k . The other components are reached by crossing the cuts, so that there is a one to one correspondence between finite sequences {l ±1 j 1 , . . . , l ±1 j m } not containing successively a l ±1 k and l ∓1 k , and components of R (here j 1 , . . . , j m ∈ {1, . . . , n} and m ∈ N). The relations (2.28) alone do not identify a component of R (as incorrectly written in [23], page 387). For example, the word l 1 l 2 l −1 Stokes matrices for (1.1), for fixed and pairwise distinct u 0 1 , . . . , u 0 n , can been expressed in terms of connection coefficients of selected solutions of (2.26). The 6 As well known, the analytic continuation, starting from the plane P η (0) , of a fundamental matrix solution of (2.26) defines a function on R(C\{u 0 1 , . . . , u 0 n }). For example, if λ is the lift of λ ∈ P η (0) to the component of R identified by the word l 1 l 2 l −1 where M j is the monodromy matrix associated with l j . explicit relations have been obtained in [4] for the generic case when all λ 1 , . . . , λ n / ∈ Z; and in [23] for the general case with no restrictions on λ 1 , . . . , λ n and A.

Selected vector solutions
The Laplace transform involves three types of vector solutions of (2.26), denoted in [23], respectively, by k (λ), * k (λ) and does not appear, since it reduces to Y k in the generic case λ k / ∈ Z). We will not describe here the * k (λ), which play mostly a technical role. Let It is proved in [23] that for every k ∈ {1, . . . , n} there are at least n − 1 independent vector solutions holomorphic at λ = u 0 k . The remaining independent solution is singular at λ = u 0 k , except for some exceptional cases possibly occurring when λ k ≤ −2 is integer. In such cases, there exist n holomorphic solutions at λ = u 0 k (such cases never occur if none of the eigenvalues of A is a negative integer). The selected vector solutions k are obtained as follows.
• If λ k ≤ −2 is integer and we are in an exceptional case when there are no singular solutions at u 0 k , namely then k is the unique analytic solution with the following normalization: • In all other cases, there is a solution with singular behaviour at λ = u 0 k . This is determined up to a multiplicative factor and the addition of an arbitrary linear combination of the remaining n − 1 regular at λ = u 0 k solutions, denoted below with reg(λ − u 0 k ). In [23], it has the following structure (2.29) The coefficients b (k) l ∈ C n are uniquely determined by the normalization. Then the selected vector solutions k are uniquely defined by 7 (2.30) In case λ k ∈ N, depending on the system, it may exceptionally happen that is the difference of two singular solutions defined on P η (0) . Here, in the notation of Remark 2.2, the function (sing) k (l k (λ)) is the value at λ ∈ P η (0) of the analytic continuation of (sing) k (λ) when passing from a prefixed component of R , in this case P η (0) , to the component associated with the sequence {l k } of only one element.
Namely, the analytic continuation for a small loop (λ − u

Connection coefficients
Above, the behaviour of k (λ) has been described at λ = u 0 k . The behaviour at any point λ = u 0 j , for j = 1, . . . , n, will be expressed by linear relations [ (2.31) 7 The singular part of (sing) is uniquely determined by the normalization, but not (sing) itself, because the analytic additive term reg(λ − u 0 k ) is an arbitrary linear combination of the remaining n − 1 independent analytic solutions. . 2 The poles u 0 j , 1 ≤ j ≤ n, of system (2.26) and plane P η (0) with branch cuts L j The above relations define the connection coefficients c jk . From the definition, we see that c kk = 1 for λ k / ∈ Z, while c kk = 0 for λ k ∈ Z. In case λ k ∈ N, if it happens that k ≡ 0, then c jk = 0 for any j = 1, .., n. Proposition 2.4 (see [4] and Propositions 3, 4 of [23]). If A has no integer eigenvalues, then (each k occupies a column) is a fundamental matrix solution of (2.26). Moreover, the matrix C := (c jk ) is invertible if and only if A has no integer eigenvalues. If A has integer eigenvalues and is fundamental, then some λ k ∈ Z.

Laplace transform and stokes matrices in terms of connection coefficients
If η (0) is admissible in the λ-plane, with respect to the fixed and pairwise distinct u 0 1 , . . . , u 0 n , then is an admissible direction (2.2) in the z-plane for system (1.1) at the fixed u = u 0 . We consider the Stokes rays of (u 0 )) as before. For some ν ∈ Z, a labelling (2.3) holds, so that In order to keep track of (2.33), we label (2.32) with ν, The connection coefficients will be labelled accordingly as c jk . Also the singular vector solutions will be labelled (sing) k (λ |ν), λ ∈ P η (0) as above.
The relation between solutions k (λ |ν) or (sing) k (λ |ν) and the columns of Y ν (z) is established in [23] for all values of λ 1 , . . . , λ n , and in [4] for non integer values only. It is given by Laplace-type integrals (Proposition 8 of [23]) Here, γ k (η (0) ) is the path coming from ∞ along the left side of the oriented L k (η (0) ), encircling u 0 k with a small loop excluding all the other poles, and going back to ∞ along the right side of L k (η (0) ).
The same as (2.34) can be defined for the cut-plane P η , with an admissible direction η satisfying and will be denoted by ν+kμ (0) (λ), and analogously for the vectors k (λ |ν + kμ (0) ) and Introduce in {1, 2, . . . , n} the ordering ≺ given by The following important results, proved in theorem 1 of [23] for all values of λ 1 , . . . , λ n , and in the seminal paper [4] in the generic case λ 1 , . . . , λ n / ∈ Z, establish the relation between Stokes matrices and connection coefficients. 8 8 The key point is the fact that (sing) k in (7.4), or equivalently k for λ 1 , . . . , λ n / ∈ Z, can be substituted by another set of vector solutions, denoted in [23] by * k (λ, u |ν) and in [4] by Y * k . The effect of the change of the branch cut from η ν+1 < η < η ν to η ν+μ+1 < η < η ν+μ , namely from η to η = η − π , yields a linear relation * where the connection matrix C + ν is expressed in terms of the connection coefficients c The same can be done for the change of branch cut from η ν+μ+1 < η < η ν+μ to η ν+2μ+1 < η < η ν+2μ (namely, η = η − π and η = η − 2π ) yielding a relation * Substituting these relations in the Laplace integrals, one proves Theorem 2.3, being S ν = C + ν and S −1 ν+μ = C − ν . See [4] and [23] where, In the above discussion, the differential systems do not depend on parameters (u is fixed). The purpose of the present paper is to extend the description of Background 2 to the case depending on deformation parameters and include coalescences in D(u c ), and then to obtain Theorem 2.2 of Background 1 in terms of an isomonodromic Laplace transform.

Equivalence of the isomonodromy deformation equations for (1.1) and (1.4)
The first step in our construction is Proposition 3.1, establishing the equivalence between strong isomonodromy deformations of systems (1.1) and (1.4), for u varying in a τ -cell of D(u c ). In the specific case of Frobenius manifolds, this fact can be deduced from Chapter 5 of [20]. Here we establish the equivalence in general terms. According to Theorem 2.
On the other hand, system (1.4) is strongly isomonodromic in D(u 0 ) by definition ( [25], Appendix A), when fundamental matrix solutions in Levelt form at each pole λ = u j , j = 1, . . . , n, have constant monodromy exponents and are related to each other by constant connection matrices (not to be confused with the connection coefficients). From [7,8,25], the necessary and sufficient condition for the deformation to be strongly isomonodromic (this can also be taken as the definition) is that (1.4) is the λ-component of a Frobenius integrable Pfaffian system with the following structure The integrability condition d P = P ∧ P is the non-normalized Schlesinger system (see Appendix A and [6][7][8]25,27,63]) Proof See Appendix B.

Schlesinger system on D(u c ) and vanishing conditions
In this section, Proposition 4.1, we holomorphically extend to D(u c ) the nonnormalized Schlesinger system associated with (1.4), when certain vanishing conditions (4.2) are satisfied. This is the second step to obtain the results of [13] by Laplace transform.
i) The vanishing relations hold if and only if Proof Let u * ∈ , so that for some i = j it occurs that u * i = u * j . Since it is an elementary computation to check the equivalence between the relation the statement on its analyticity is straightforward. For completeness, we also state the following By holomorphy of A(u) on D(u c ), the r.h.s is well defined, so that also the l.h.s. must be holomorphic on D(u c ). From (4.6) we proceed as in Remark 2.1, concluding that

Proposition 4.1 Consider a Frobenius integrable Pfaffian system
The proof can be done also with an argument similar to Remark 6.1.

Selected vector solutions depending on parameters u ∈ D(u c ), Theorem 5.1
In this section, we state one main result of the paper, Theorem 5.1, introducing the isomonodromic analogue of the selected and singular vector solutions (2.30) and (2.29). This is the third step required to obtain the results of [13] by Laplace transform. Preliminarily, we characterize the radius 0 > 0 of D(u c ) in (2.1). The coalescence point u c = (u c 1 , . . . , u c n ) contains s < n distinct values, say λ 1 , . . . , λ s , with algebraic multiplicities p 1 , …, p s , respectively ( p 1 + · · · + p s = n). Suppose that arg z = τ is a direction admissible at u c , as defined in (2.20), and let η = 3π/2 − τ be the corresponding admissible direction in the λ-plane, where we draw parallel half lines L 1 = L 1 (η), …, L s = L s (η) issuing from λ 1 , …, λ s , respectively, with direction η, as in Fig. 3. Let 2δ αβ := distance between L α and L β , The bound (5.1) was introduced in [13] in order to prove Theorem 2.2 in Background 1. It implies properties of the Stokes rays as u varies in D(u c ), described later in Sect. 7. Let be the disc centered a λ α and radius 0 . If u j is such that u c j = λ α , the bound (5.1) implies that u j remains in D α as u varies in D(u c ). Clearly, D α ∩ D β = ∅. Fig. 3 The figure represents the half lines L α , L β , etc, for α, β, . . . ∈ {1, . . . , s}, in direction η = 3π/2−τ , the discs centred at the coordinates λ 1 , . . . , λ s of the coalescence point u c , and the distances δ αβ . Also two points u i , u j are represented, such that u c i = u c j = λ δ for some δ ∈ {1, . . . , s}. Important: now η refers to u c , differently from Sect. 2.2 and Fig. 2 The Stokes rays of (u c ) can be labeled as in (2.21). For a certain ν ∈ Z we have For each u ∈ D(u c ), let P η = P η (u) be the λ-plane with branch cuts L 1 = L 1 (η), …, L n = L n (η) issuing from u 1 , . . . , u n and the choice of the logarithms ln(λ − u k ) = ln |λ − u k | + i arg(λ − u k ), given by We define the domain (notation× inspired by [33]) are equivalent to Frobenius integrability on the whole D(u c ). With this in mind, we state the following Theorem 5.1 Consider a Pfaffian system,  λ, u |ν), . . . , n (λ, u |ν). Each k (λ, u |ν) is uniquely identified by the local behaviour below for λ ∈ D α , where α is such that u c k = λ α . The label ν keeps track of (5.2).

4)
where ψ k (λ, u |ν) is holomorphic on D α ×D(u c ) and is represented by a uniformly convergent Taylor expansion with holomorphic on D(u c ) coefficients:

5)
The following normalization uniquely identifies k .
is holomorphic on D α × D(u c ), the Taylor expansion being uniformly convergent with holomorphic coefficients d The isolated singularities of k (λ, u |ν), if any, are located at λ = u j with u c j = λ β , β = α, and at λ = u k only in case λ k ∈ C\Z. For i = j such that u c i = u c j , i (λ, u |ν) and j (λ, u |ν) are either linearly independent, or at least one of them is identically zero (identity to zero may occur only for λ i or λ j belonging to N) (λ, u |ν) is a solution with an isolated singularity at λ = u k , whose singular behaviour is uniquely characterized as follows. 11 Let D α be identified by λ α = u c k . • Forλ k ∈ C\Z [algebraic or logarithmic branch-point], where * m =k is over all m such that u m ∈ D α and λ m ∈ Z − . The vector function In particular, for λ k ≤ −2, depending on the system, it may happen that there is no solution with singularity in D α , so that the Taylor expansion being uniformly convergent and the coefficients b Let i, j be such that u c i = u c j . Then (u) will be useful later.

Remark 5.2 For λ k /
∈ Z − , the singular solution (sing) k is unique, identified by its singular behaviour at λ = u k and the normalization (5.5)-(5.6) when λ k ∈ C\Z, or by the normalization (5.11) when λ k ∈ N. For λ k ∈ Z − , a singular solution in (5.8) is not unique, but its singular behaviour (5.9) at λ = u k is uniquely fixed by the normalization (5.5)-(5.6). There is a freedom due to the choice of the coefficients r m and of φ k in (5.8). See also Remark 6.3.
The singular behaviour of k at λ = u j is expressed by connection coefficients.

Definition 5.1
The connection coefficients are defined by The uniqueness of the singular behaviour of (sing) j at λ = u j implies that the c jk are uniquely defined. From the definition, we see that
←− row j is possibly non zero .

The exponents T (l) and R ( j) satisfy the following commutation relations
By analytic continuation, ( p 1 ) (λ, u) defines an analytic function on the universal covering of P η (u)×D(u c ). Another representation of (6.9) will be given in (6.24).

Proof
for certain matrices A j which are simultaneous triangular forms of B 1 (u c ), . . . , B p 1 (u c ). While in [63] a lower triangular form is considered, we equivalently use the upper triangular one. The matrices Q j will be described below. The matrix The constant matrix coefficients V k , W k 1 ,...,k p 1 can be determined [63] from the constant matrix coefficients P i,k in the Taylor expansion 15 of the P j (x) and P j (x). Recall that x j = λ − u j , 1 ≤ j ≤ p 1 , and x n+1 = λ − λ 1 . Moreover, for Therefore, the matrices appearing in the statement are G ( p 1 ) : We show that the exponents A j and Q j are, respectively, T ( j) in (6.6)-(6.7) and R ( j) in (6.10)-(6.11)-(6.12). According to [63] (see theorems 2 and 5), the matrix function G ( p 1 ) · U ( p 1 ) (λ, u) in (6.9) provides the gauge transformation = G ( p 1 ) · U ( p 1 ) (λ, u)Z ≡ in notation of [63] U 0 U (x)Z , which brings (6.3) to the reduced form (being "reduced" is defined in [63]) where the notation k = (k 1 , . . . , k p 1 ) > 0 means at least one k l > 0. From [63], we have the following.
and analogous for P j (x) • The Q k, j satisfy diag(Q k, j ) = 0, while the entry (α, β) for α = β satisfies Taking into account the particular structure (6.6)-(6.7), the above condition can be satisfied only for This can occur only for j = q 1 + 1, . . . , p 1 . Thus In conclusion, the reduced form turns out to be Its integrability implies the commutation relations. Indeed, the compatibility Keeping into account that R (1) = · · · = R (q 1 ) = 0, the above holds if and only if (6.13)-(6.14) hold. The last to be checked is that a fundamental matrix of (6.19) is Z (x) in (6.15), namely It suffices to verify this by differentiating Z (x), keeping into account the commutation relations (6.13)-(6.14) and the formula
Now, recalling that k i = |λ i + 1| and (6.16)-(6.17), we see that as we wanted to prove. Finally, the fact that ( p 1 ) (λ, u) has analytic continuation on the universal covering of P η (u)×D(u c ) follows from general results in the theory of linear Pfaffian systems [28,32,63].
This new definitions allow to treat together the case λ j ∈ −N−2 and the case λ j = −1.
Proof It is an immediate consequence of the commutation relations being satisfied, that the representation (6.9) for ( p 1 ) still holds with the definition (6.20)-(6.21).
The commutation relations impose a simplification on the structure of the matrices R ( j) . Let the new convention (6.20)-(6.21) be used. The relations [T (i) , R ( j) ] = 0 for i = 1, . . . , p 1 and j = q 1 + 1, . . . , p 1 , j = i, imply the vanishing of the first p 1 non-trivial entries of R ( j) , so that (by (6.11), (6.12) and (6.21)), In particular, if λ j = −1 and R ( j) is (6.21), all the above conditions can be satisfied, provided that we take m j ≥ p 1 + 1, as we have agreed from the beginning.

Selected vector solutions 9 i
Remark on notations For the sake of the proof, it is convenient to use a slightly different notation with respect to the statement of Theorem 5.1. The identifications between objects in the proof and objects in the statement is We will construct selected vector solutions of Theorem 5.1 from suitable linear combinations of columns of the fundamental matrix ( p 1 ) in (6.24). The ith column of an n × n matrix M is M · e i (rows by columns multiplication), where e i is the standard unit basic vector in C n . From (6.22)-(6.23), and (6.26)-(6.25)-(6.27), we receive (6.28) which is holomorphic for (λ, u) ∈ D 1 × D(u c ). For i = 1, . . . , p 1 , we define vector valued functions Hence, the ith column of ( p 1 ) (λ, u) is  Proof For i = 1, . . . , q 1 + c 1 , (6.35) is just (6.32), so it is a vector solution of (5.3).

Singular solutions 9 (sing) i
Using the previous results, we define singular vector solutions of the Pfaffian system.

Remark 6.3
The definition in (i) contains the freedom of choosing k ∈ {p 1 +1, . . . , n}, which changes ϕ k (λ, u) and the ratios r Whatever is the choice of k, provided that r (i) k = 0, the behaviour at λ = u i of the corresponding (sing) i is always (6.36), so it is uniquely fixed if we fix the normalization of i (λ, u).
As a consequence of the above definitions and Sect. 6.2, we receive the following   = u i , i = 1, . . . , p 1

and completion of the proof
In order to proceed in the proof, and in view of the Laplace transform to come, we need local behaviour at λ = u i .

Lemma 6.3
The following Taylor expansion holds at λ = u i , with coefficients d , so it is holomorphic on D 1 × D(u c ). From this we conclude.
The coefficients d (i) l (u) will be fixed by a chosen normalization for ϕ i in (6.33), as in the following lemma.

Lemma 6.4
The following Taylor expansions hold at λ = u i , uniformly convergent for u ∈ D(u c ).
with certain vector coefficients b In particular, the leading term is constant, and will be chosen as follows Proof That the above convergent expansions must hold follows from the definitions. Work is required to prove that the leading term is f i e i , with f i ∈ C\{0}. From definitions (6.29)-(6.30), the leading term must coincide with the leading term of the expansion at λ = u i of the ith column G ( p 1 ) U (λ, u) · e i , for i = 1, . . . , p 1 . To evaluate it, observe that the solution ( p 1 ) (λ, u), restricted to a polydisc D(u 0 ) contained in a τ -cell of D(u c ), is a fundamental matrix solution of the Fuchsian system (1.4) in the Levelt form (6.38) at λ = u i , i = 1, . . . , p 1 . Indeed, by (6.23) it can be written as where it is understood that R (i) = 0 if i = 1, . . . , q 1 . We have The matrix G (i; p 1 ) (u) is holomorphically invertible if restricted to a polydisc D(u 0 ) contained in a τ -cell, but it is branched at the coalescence locus on the whole D(u c ). We reach our goal if we show that the ith column G (i; p 1 ) (u) · e i is constant in D(u c ). First, it follows from (6.38) and the standard isomonodromic theory of [33] that G (i; p 1 ) (u) holomorphically in D(u 0 ) reduces B i (u) to the diagonal form T (i) , when λ i = −1, or to non-diagonal Jordan form (6.21) when λ i = −1 For this reason, the ith row is proportional to the eigenvector e i of B i (u) relative to the eigenvalue −λ i − 1. Namely, for some scalar function f i (u), This is obvious for λ i = −1, namely for diagonalizable B i . If λ i = −1, the eigenvalue 0 of B i appearing in J (i) at entry (i, i) is associated with the eigenvector f i (u) e i . Moreover, for every invertible matrix G = [ * | · · · | * | e i | * | · · · | * ], where e i occupies the kth column, then G −1 B i (u)G is zero everywhere, except for the kth row. Now, since R (i) = J (i) has only one non-zero entry on the ith row, it follows that the eigenvector f i (u) e i must occupy the ith column of G (i; p 1 ) (u).
• f i (u) is holomorphic on D(u c ). Indeed, by (6.28), From (2.18) and (4.3), the ith column of B j u j − u i + ω j is null. Hence, Moreover, summing the Eq. (6.39), we get n j=1 ∂ j G (i; p 1 ) = 0. Thus, G (i; p 1 ) · e i is constant on D(u 0 ), and being holomorphic on D(u c ), it is constant on D(u c ). The choice (6.37) will be made.
The above obtained expansions for the i and (sing) i and ϕ i prove Theorem 5.1 for i = 1, . . . , p 1 , with some obvious identifications between objects in the proof and objects in the statement, namely

Proof of Proposition 5.1
Proof For simplicity, we omit ν in the connection coefficients c (ν) jk in (5.12)-(5.13). It follows from the very definitions of the k and In order to prove independence of u, we express the monodromy of (λ, u) := [ 1 (λ, u) | · · · | n (λ, u)], in terms of the connection coefficients. From the definition, we have (using the notations in the statement of Theorem 5.1) ∈ and a small loop (λ − u k ) → (λ − u k )e 2πi we obtain from Theorem 5.1 k (λ, u) −→ k (λ, u)e −2πiλ k , which includes also the case λ k ∈ Z, with e −2πiλ k = 1.
For a small loop (λ − u j ) → (λ − u j )e 2πi , j = k, from Theorem 5.1 and (6.40) we obtain Therefore, for u / ∈ and a small loop γ k : (λ − u k ) → (λ − u k )e 2πi not encircling other points u j (we denote the loop by λ → γ k λ), we receive We proceed by first analyzing the generic case, and then the general case. Generic case. Suppose that A(u) has no integer eigenvalues (recall that eigenvalues do not depend on u). Let us fix u in a τ -cell. By Proposition 2.4, (λ, u) is a fundamental matrix solution of (1.4) for the fixed u, and C = (c jk ) is invertible. Thus The above makes sense for every u in the considered τ -cell, being (λ, u) invertible at such an u. But (λ, u) and (γ k λ, u) are holomorphic on P η (u)×D(u c ), so that the matrix M k (u) is holomorphic on the τ -cell. Repeating the above argument for another τ -cell, we conclude that M k (u) is holomorphic on each τ -cell. Now, on a τ -cell, we have and at the same time The two expressions are equal if and only if d M k = 0, because (λ, u) is invertible on a τ -cell. Recall that τ -cells are disconnected from each other, so that separately on each cell, M k is constant, and so the connection coefficients are constant separately on each cell.
We further suppose that none of the λ j is integer. In this case, (sing) j = j for all j = 1, . . . , n, so that from (6.40) for u c k = u c j (otherwise c jk = 0 and there is nothing to prove) k (λ, u) = λ→u j j (λ, u)c jk + reg(λ − u j ). (6.41) Using the labelling (6.1)-(6.2), from the Proof of Theorem 6.1 we have the fundamental matrix solution and in general at each λ α , α = 1, . . . , s (with α−1 j=1 p j = 0 for α = 1) we have and the ψ m (λ, u) and ϕ (α) r (λ, u) are holomorphic functions in the corresponding D α × D(u c ). The above allows us to explicitly rewrite (6.41), for j such that u c j = λ α , as 42) for suitable constant coefficients h r . Here one of the c mk is c jk of (6.41).
General case of any A(u). If some of the diagonal entries λ 1 , . . . , λ n of A are integers, or some eigenvalues are integers, there exists a sufficiently small γ 0 > 0 such that, for any 0 < γ < γ 0 , A − γ I has diagonal non-integer entries λ 1 − γ, . . . , λ n − γ and no integer eigenvalues. Take such a γ 0 , and for any 0 < γ < γ 0 consider Lemma 6.5 The above system (6.44) is strongly isomonodromic in D(u 0 ) contained in a τ -cell, and λ-component of the integrable Pfaffian system For u ∈ D(u 0 ) contained in a τ -cell, we write the unique formal solution The crucial point is that F(z, u) is the same as (2.5), so all the F k (u) are independent of γ . The fundamental matrix solutions are uniquely defined by their asymptotics γ Y F (z, u) in S ν (D(u 0 )). Their Stokes matrices do not depend on γ because The system (6.47) is thus strongly isomonodromic. By Proposition 3.1 we conclude.
For system (6.45) the results already proved in the generic case hold. Therefore, the connection coefficients c (6.49) are constant on D(u c ). They depend on γ , but not on u ∈ D(u c ).

Remark 6.4
It is explained in section 8 of [23] what is the relation between (sing) k and γ k , by means of their primitives, and that in general both lim γ →0 γ k and lim γ →0 c (ν) jk [γ ] are divergent. Now, we invoke Proposition 10 of [23], which holds with no assumptions on eigenvalues and diagonal entries of A(u). 18 This result, adapted to our case, reads as follows. Proposition 6.3 Let u be fixed in a τ -cell. Let γ 0 > 0 be small enough such that for any 0 < γ < γ 0 the matrix A − γ I has no integer eigenvalues, and its diagonal part has no integer entries. 19 Then, the following equalities hold , if k ≺ j; (6.50) where the ordering relation j ≺ k means, for the fixed u, that (z(u j − u k )) < 0 for arg z = τ = 3π/2 − η satisfying (5.2).
We use Proposition 6.3 to conclude the proof of Proposition 5.1 in the general case. Indeed, the proposition is already proved in the generic case, so it holds for the c (ν) jk [γ ]. Therefore, they are constant on the whole D(u c ). Equalities (6.50) hold at any fixed u in τ -cell, so that each c (ν) jk is constant on a τ -cell, and such constant is the same in each τ -cell. With a slight variation of η in (η ν+1 , η ν ), equalities (6.50) hold also at the crossing locus X (τ ). They analytically extend at . Let τ be the chosen direction in the z-plane admissible at u c , and η = 3π/2 − τ in the λ-plane. The Stokes rays of (u c ) will be labelled as in (2.21), so that (5.2) holds for a certain ν ∈ Z. We define the sectors S ν = {z ∈ R(C\{0}) such that τ ν − π < arg z < τ ν+1 }. (7.1) If u only varies in D(u 0 ) contained in a τ -cell, then none of the Stokes rays associated with (u) crosses arg z = τ mod π . If u varies in D(u c ), some Stokes rays associated with (u) necessarily cross arg z = τ mod π (see Sect. 2.1.2). Consider the subset of the set of Stokes rays satisfying (z(u j − u k )) = 0, z ∈ R, associated with pairs (u j , u k ) such that u j ∈ D α and u k ∈ D β , α = β, namely u c j = u c k . Following [13], we denote this subset by R(u). If u varies in D(u c ) and 0 satisfies (5.1), the rays in R(u) continuously rotate, but never cross the admissible rays arg z = τ + hπ , where The above allows to define S ν+hμ (u) to be the unique sector containing S τ + (h − 1)π, τ + hπ and extending up to the nearest Stokes rays in R(u). Then, let 3) It has angular amplitude greater than π . The reason for the labeling is that S ν+hμ (u c ) = S ν+hμ in (7.1).
with asymptotic behaviour, uniform in u ∈ D(u c ), given by the formal solution Y F (z, u) = F(z, u)z B e z . The coefficients F l (u) are holomorphic in D(u c ). The explicit expression of their columns is (7.12), (7.13), (7.15) [or (7.16)] and (7.17).

(3) Stokes matrices defined by
where the relation j ≺ k is defined for j = k such that u c j = u c k and means that (z(u c j − u c k )) < 0 when arg z = τ . Remark 7. 1 The above (7.9) generalizes Theorem 2.3 in the presence of isomonodromic deformation parameters, including coalescences. Notice that the ordering relation ≺ here is referred to u c , while in Theorem 2.3 it refers to u 0 .
• Construction of Y k (z, u |ν). We have (sing) k (λ, u| ν) = k (λ, u| ν) and (7.4) is Since k (λ, u |ν) grows at infinity no faster than some power of λ, the integral converges in a sector of amplitude at most π . Now, k (λ, u |ν) satisfies Theorem 5.1, hence if u varies in D(u c ) the following facts hold.
(1) k (λ, u |ν) is branched at λ = u k and possibly at other poles u l such that u c l = u c k . (2) k (λ, u |ν) is holomorphic at all λ = u j such that u c j = u c k , j = k. It follows from (1) to (2) that the path of integration can be modified: for α such that u c k = λ α , we have e zλ k (λ, u |ν)dλ, where α (η) is the path which comes from ∞ in direction η − π , encircles λ α along ∂D α anti-clockwise and goes to ∞ in direction η. This path encloses all the u j such that u c j = λ α , end excludes the others. See Fig. 4. We conclude that u can vary in D(u c ) and the integral (7.11) converges for z in the sector defining Y k (z, u |ν) as a holomorphic function of (z, u) ∈ S(η) × D(u c ). Now, if u varies in D(u c ) and 0 satisfies (5.1) none of the vectors u i − u j such that u c i = λ α and u c j = λ β , 1 ≤ α = β ≤ s, cross a direction η mod π , for every η ν+1 < η < η ν . Due to 1. and 2. above, a vector function k (λ, u |ν) is well defined in P η and Pη for any η ν+1 < η <η < η ν , and so on P η ∪ Pη. Therefore, the integral in (7.11) satisfies namely one is the analytic continuation of the other, so defining the function Y k (z, u |ν) as analytic on S ν × D(u c ), where coincides with (7.3) (with h = 0). Finally, notice that e λz (λ − ) k (λ, u |ν) (α) = 0, due to the exponential factor.
• Asymptotic behaviour. From (5.4)-(5.5), we write (7.11) as We conclude that Y k (z, u |ν) is analytic on S ν × D(u c ). Moreover, e λz (λ − ) Keeping into account that the integral along α can be interchanged with that along γ k , it follows that We conclude, by the standard evaluation of the remainder analogous to R N (z) considered before, and the variation of η in the range (η ν+1 , η ν ), that 20 We conclude analogously to previous cases that Y k (z, u |ν) is analytic in S ν ×D(u c ). It is a solution of (1.1), by (2.25), because k (λ, u |ν) is analytic at λ = u k and behaves as in (5.4)-(5.5), so that e λz (λI − (u)) k (λ, u |ν) • Asymptotic behaviour. We have, from (5.4)-(5.5), We integrate term by term in order to obtain the asymptotic expansion (the remainder for the truncated series is evaluate in standard way, as R N (z) above). For the integration, we use

Fundamental matrix solutions
The vector solutions Y k (z, u |ν) constructed above can be arranged as columns of the matrix Y ν (z, u) := Y k (z, u |ν) · · · Y n (z, u |ν) , which thus solves system (1.1). From the general theory of differential systems, it admits analytic continuation as analytic matrix valued function on R(C\{0}) × D(u c ).
Letting B = diag A = diag(λ 1 , . . . , λ n ), the asymptotic expansions obtained above are summarized as Therefore, the coefficients F l (u) of the formal solution Y F (z, u) = F(z, u)z B e (u)z are holomorphic in D(u c ). Moreover, the leading term is the identity I , which implies that Y ν (z, u) is a fundamental matrix solution.
Stokes matrices are defined by (7.7). Thus, S ν+hμ (u) = Y ν+hμ (z, u) −1 Y ν+(h+1)μ (z, u) is holomorphic in D(u c ). Let us consider the relations for h = 0, 1: Let u be fixed in a τ -cell, so that has distinct eigenvalues. From Theorem 2.3 at the fixed u we receive Here, for j = k the ordering relation j ≺ k ⇐⇒ (z(u j − u k ))| arg z=τ < 0 is well defined for every u in the τ -cell, because no Stokes rays (z(u j − u k )) = 0 cross arg z = τ as u varies in the τ -cell. The relation j ≺ k may change to j k when passing from one τ -cell to another only for a pair u j , u k such that u c j = u c k . This is due to the choice of 0 as in (5.1). On the other hand, c (ν) jk = 0 whenever u c j = u c k . This means that (7.9) is true at every fixed u in every τ -cell, with ordering relation j ≺ k defined for j = k such that u c j = u c k , namely (z(u c j − u c k )) < 0 when arg z = τ . Since the S ν+hμ are holomorphic in D(u c ) and the c (ν) jk are constant in D(u c ), we conclude that Stokes matrices are constant in D(u c ) and hence (7.9) holds in D(u c ). The vanishing conditions (7.8) follow from the vanishing conditions (5.14) for the connection coefficients, plus the fact that we can generate all the S ν+hμ from the formula S ν+2μ = e −2πi B S ν e 2πi B .

Non-uniqueness at u = u c of the formal solution
By Laplace transform, we prove Corollary 2.1 in Background 1, asserting that system (2.19) has unique formal solution if and only if the constant diagonal entries of A(u) satisfy the partial non-resonance Otherwise, the Laplace transform will be proved to generate a family of formal solu-tionsY whose coefficientsF l depend on a finite number of arbitrary parameters.
Due to the strategy of Sect. 6.6, it will suffice to consider the generic case when all λ 1 , . . . , λ n / ∈ Z and A has no integer eigenvalues. Indeed, if this is not the case, the gauge transformation (6.46) relates a formal solution γ Y F to Y F at any point u, through (6.48), so that the coefficients F l of a formal expansion do not depend on γ . We are interested in these coefficients.
Consider system (1.4) under the assumptions that it is (strongly) isomonodromic in D(u c ), so that (A) i j (u c ) = 0 for u c i = u c j . For simplicity, we order the eigenvalues as in (6.1)-(6.2). Since Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Proof For every i ∈ {1, . . . , n}, the Pfaffian system (3.2) can be rewritten as

Appendix A. non-normalized Schlesinger system
We study λ − u i → 0, while u j − u i = 0 for j = i in D(u 0 ). In new variables λ = λ, y i = λ − u i , y j = u j − u i , j = i, P is rewritten in the following way (which defines the matrices A j (y)) The only singular term at y i = 0 is B i /y i in A i (y). The components relative to dy 1 , . . . , dy n of d P = P ∧ P are For k = i and l = i, from (9.1), we receive where reg(y i ) stands for an analytic term at y i = 0. We expand the above in Taylor series at y i = 0. The singular term (the residue at y i = 0) is The above gives the non-normalized Schlesinger Eqs. (3.4)-(3.5), because If we write the components of d P = P ∧ P referring to dy l ad dλ, and we substitute into them (9.3)-(9.4), we receive (3.3), namely ∂ l γ k − ∂ k γ l = γ l γ k − γ k γ l . Proof We must show that there exists a holomorphically invertible G (i) (u) on D(u 0 ) such that (G (i) ) −1 B i G (i) is a constant Jordan form. The conditions (9.1) for k, l = i can be evaluated at y i = 0, and become Hence, the following Pfaffian system is Frobenius integrable Using the chain rule as in (9.3), we receive (6.8) Notice that for both ϕ(u) = B i (u) and ϕ(u) = G(u) we have n k=1 ∂ϕ ∂u k = 0 ⇒ ϕ(u) = ϕ(u 1 − u i , . . . , u n − u i ). (9.7) We can take a solution G(u) which holomorphically reduces B i to Jordan form. Indeed Therefore, keeping into account (9.7), we see that B i := G −1 (u)B i (u)G(u)) is independent of u. Thus, there exists a constant matrix G such that G −1 B i G is a constant Jordan form, and G (i) (u) := G(u)G realizes the holomorphic "Jordanization". The above arguments are standard, see for example [28].
If the B i (u) are holomorphic on D(u c ) and the vanishing conditions (4.1) hold, the coefficients of the Pfaffian system (6.39) are holomorphic on D(u c ), so that G (i) (u) extends holomorphically there, and Corollary 9.1 holds on D(u c ). . We multiply these equations to the left by E k , with k = i. We receive The l.h.s. is E k ∂ i A = ∂ i B k . The r.h.s. is In conclusion The only terms we need to evaluate are In the second line we have used E i E k = E k E i = E i B k = 0, for i = k, and E 2 i = E i . Now, observe that E k F 1 E i has zero entries, except for the entry (k, i), which is (F 1 ) ki = (A) ki /(u i − u k ). This implies that In conclusion, we have proved that (3.1) implies (3.4). On the other hand (3.4)- (3.5) are equivalent to the system given by (3.4)  which are exactly (3.1) if B k = E k A. Finally, notice that (3.3), here with γ j = ω j , is the integrability condition on D(u 0 ) of dG = n j=1 ω j (u)du j G. On the other hand, it is a computation to see that (3.1) implies the the same conditions. Conversely, let system (1.4) be strongly isomonodromic, so that the integrability conditions (3.3)-(3.5) hold. Firstly, we show that (3.4)-(3.5) imply a Pfaffian system for A of type (3.1). To this end, we sum (3.4) and (3.5): Using B k = −E k (A + I ) and k E k = I , the above becomes The above equations imply that Thus, G(u) also puts A in constant Jordan form, so that 22 where G (0) is in (2.12).
In particular, G (0) satisfies (10.1). The second part of the statement of Proposition 2.3 (Prop. 19.2 of [13]) is now easily proved. Indeed, if A(u) = G (0) (u)J (G (0) ) −1 holomorphically on D(u c ), where J is Jordan, then G (0) satisfies (10.1) on D(u 0 ) (and J is constant). Since G (0) (u) is holomorphic on D(u c ), the ω j must be as well, so that the vanishing conditions (2.22) must hold. Conversely, if A is holomorphic on on D(u 0 ) and satisfies the vanishing conditions (2.22) (or, more weakly, if d A = j [ω j , A]du j on D\ , which automatically implies (2.22)-see Remark 2.1), then dG = j ω j du j G is integrable with holomorphic coefficients on D(u 0 ), and admits a fundamental matrix solution that can be chosen so that (G (0) ) −1 AG (0) (u) = J (the proof is as done before on D(u 0 )).