Flat structure on the space of isomonodromic deformations

Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as Frobenius manifold structure. In this paper, we study isomonodromic deformations of an Okubo system, which is a special kind of systems of linear differential equations. We show that the space of independent variables of such isomonodromic deformations can be equipped with a Saito structure (without a metric), which was introduced by C. Sabbah as a generalization of Frobenius manifold. As its consequence, we introduce flat basic invariants of well-generated finite complex reflection groups and give explicit descriptions of Saito structures (without metrics) obtained from algebraic solutions to the sixth Painlev\'{e} equation.


Introduction
At the end of 1970's, K. Saito introduced the notion of flat structure in order to study the structure of universal unfolding of isolated hypersurface singularities. Independently the WDVV equation (Witten-Dijkgraaf-Verlinde-Verlinde equation) arises from the 2D topological field theory [15,62]. B. Dubrovin unified both the flat structure formulated by K. Saito and the WDVV equation as Frobenius manifold structure. Dubrovin not only formulated the notion of Frobenius manifold but also studied its relationship with isomonodromic deformations of linear differential equations with certain symmetries. Particularly he derived a one-parameter family of Painlevé VI equation from three-dimensional massive (i.e. regular semisimple) Frobenius manifolds. Since then, there are several generalizations of Frobenius manifolds such as F -manifold by C. Hertling and Y. Manin [22,21] and Saito structure (without metric) by C. Sabbah [51]. Concerning the relationship with the Painlevé equation, A. Arsie and P. Lorenzoni [1,43] showed that three-dimensional regular semisimple bi-flat F -manifolds are parameterized by generic solutions to the (fullparameter) Painlevé VI equation, which is regarded as an extension of Dubrovin's result. Furthermore Arsie-Lorenzoni [2] showed that three-dimensional regular non-semisimple bi-flat F -manifolds are parameterized by generic solutions to the Painlevé V and IV equations. (Recently it was proved in [3,41] that Arsie-Lorenzoni's bi-flat F -manifold is equivalent to Sabbah's Saito structure (without metric).) The theory of linear differential equations on a complex domain is a classical branch of mathematics. In recent years there has been a great progress in this branch. One of its turning points is the introduction of the notions of middle convolution and rigidity index by N. M. Katz [33]. With the help of his idea, T. Oshima developed a classification theory of Fuchsian differential equations in terms of their rigidity indices and spectral types [49,50,48]. ( [19] is a nice introductory text on the "Katz-Oshima theory".) Okubo systems play a central role in these developments (cf. [13,14,50,64]): a matrix system of linear differential equations with the form where T, B ∞ are constant square matrices, is said to be an Okubo system if T is diagonalizable. Particularly, any Okubo system is Fuchsian. One important feature of an Okubo system is that it keeps its form under the operation of the Euler transformation (cf. Remark 2.1): Okubo system was introduced with the following motivations: • A system of differential equations in Birkhoff normal form (which has an irregular singularity of Poincaré rank one at 0 and a regular singularity at ∞ on P 1 (C)) is transformed into an Okubo system by a Fourier-Laplace transformation and thus the study on Stokes matrices of a Birkhoff normal form can be reduced to that on connection matrices of the corresponding Okubo system ( [4,55]).
• Okubo system is one of generalizations of Gauss hypergeometric equation and one may expect that it would provide new special functions possessing rich properties ( [45,63,64]).
Let us consider a regular semisimple Saito structure (without metric). (See Section 4 for the definition and properties of Saito structure (without metric). In the sequel, we abbreviate Saito structure (without metric) to Saito structure for brevity.) It is known that there are following two types of meromorphic connections associated with a Saito structure (cf. Remarks 4.2 and 4.3): (i) One induces a universal integrable deformation of a Birkhoff normal form.
(ii) The other induces a universal integrable deformation of an Okubo system.
These two meromorphic connections are called in [21] (in the case of a Frobenius manifold) the first structure connection and the second structure connection respectively and mutually equivalent since they are transformed to each other by Fourier-Laplace transformations. The first connection (i) is used in many preceding literatures (e.g. [16,51]). However in this paper we use the second connection (ii) because results in the recent developments on linear differential equations mentioned above can be exploited. On this standpoint, a regular semisimple Saito structure yields a universal integrable deformation of a regular Okubo system. In this paper, we show that the opposite of this statement is almost always true. Namely, for a universal integrable deformation of a regular Okubo system satisfying some generic condition, there exists a Saito structure which yields the given universal integral deformation as the second connection (ii) (Theorem 5.4). A rough picture of the result in this paper is regular semisimple Saito structures (2) ⇐⇒ universal integrable deformations of regular Okubo systems.
Here it should be remarked that a universal integrable deformation of a regular Okubo system turns out to be an isomonodromic deformation (cf. Remark 2.2). In this paper, the following results are obtained as consequences of (2): (I) introduction of flat coordinates on the orbit spaces of well-generated finite complex reflection groups (Theorem 6.2), (II) correspondence between solutions satisfying certain semisimplicity condition to the three-dimensional extended WDVV equation and generic solutions to the Painlevé VI equation (Corollary 5.5), (III) explicit descriptions of potential vector fields corresponding to algebraic solutions to the Painlevé VI equation (Section 7).
K. Saito and his collaborators [53,54] defined and constructed flat coordinates on the orbit spaces of finite real reflection groups. To extend them to finite (non-real) complex reflection groups has been a long-standing problem. (I) provides an answer to this problem for well-generated complex reflection groups (see also [29]). Recently this result has been crucially used to prove the freeness of multi-reflection arrangements of complex reflection groups in [25]. (II) is essentially equivalent to A. Arsie and P. Lorenzoni's result in [1,43], however the argument based on the picture (2) makes clear the relationship between Saito structures and the theory of isomonodromic deformations. (III) provides many concrete examples of three-dimensional algebraic Saito structures that are not Frobenius manifolds, which would be the first step toward classification of three-dimensional algebraic Saito structures and/or algebraic F -manifolds (cf. [21]), see also [29,30,31]. (In this paper, the term "flat structure" stands for the same meaning as "Saito structure" because it is a natural extension of K. Saito's flat structure, which would be justified by the result (I) above.) This paper is constructed as follows. In Section 2, we start with an Okubo system. Then we introduce the notion of Okubo system in several variables as an integrable Pfaffian system extending the Okubo system (Definition 2. 3). An Okubo system in several variables is equivalent to an isomonodromic deformation of an Okubo system (Remark 2.2). We see that an Okubo system in several variables is a universal integrable deformation of an Okubo system, which is uniquely determined up to changes of independent variables by the given Okubo system (Remark 2.3). In Section 3, we study the structure of logarithmic vector fields along a divisor defined by a monic polynomial of degree n h(x) = h(x ′ , x n ) = x n n − s 1 (x ′ )x n−1 n + · · · + (−1) n s n (x ′ ) where x ′ = (x 1 , · · · , x n−1 ) and each s i (x ′ ) is holomorphic with respect to x ′ , which appears as the defining equation of the singular locus of an Okubo system in several variables. In K. Saito's construction of flat structures on the orbit spaces of real reflection groups, the fact that the discriminants of real reflection groups have the form of (3) was crucially used (cf. [53]). The results in this section are used in order to generalize Saito's construction in Section 6. In Section 4, we review the general theory of Saito structure. Particularly we introduce an extension of the WDVV equation (Proposition 4.8, Definition 4.9), which is not mentioned in [51] explicitly. A solution to the extended WDVV equation is called a potential vector field, which completely describes a Saito structure (Proposition 4.10). This is an extension to Saito structure of the fact that a Frobenius manifold is completely described by its prepotential, which is a solution to the WDVV equation. In Section 5, we give a precise statement of the picture (2). Namely, we show that an Okubo system in several variables arises from a Saito structure if and only if its flat coordinate is non-degenerate in some sense (Theorem 5.4). For uniqueness of the Saito structure corresponding to an Okubo system in several variables, see Remark 5.1. Combining this result and the argument in Appendix A, we see that there is a correspondence between solutions satisfying certain semisimplicity condition to the three-dimensional extended WDVV equation and generic solutions to the (full-parameter) Painlevé VI equation (Corollary 5.5). In Section 6, we treat a problem on the existence of a flat coordinate system on the orbit space of an irreducible complex reflection group. In the case of a real reflection group, K. Saito [53] (see also [54]) proved the existence of a flat coordinate system on the orbit space based on the existence of a flat invariant metric on the standard representation space. In this paper, instead of the flat invariant metric, we construct a special type of Okubo systems in several variables called G-quotient system, whose fundamental system of solutions consists of derivatives by logarithmic vector fields of linear coordinates on the standard representation space and its monodromy group is isomorphic to the finite complex reflection group G (Theorem 6.2). Then as a consequence of the picture (2), we have a Saito structure on the orbit space corresponding to the G-quotient system. Actually, our definition of flat coordinates coincides with that of Saito when it is restricted to real reflection groups. Here it is remarkable that our proof of Theorem 6.2 is constructive i.e. it contains an algorithm of explicit computation on the flat generator system of G-invariants and the potential vector field for a given well-generated complex reflection group G. Besides, it is proved that the potential vector field corresponding to a well-generated complex reflection group has polynomial entries (Corollary 6.8). In Section 7, we treat algebraic solutions to the Painlevé VI equation. Algebraic solutions to the Painlevé VI equation were studied and constructed by many authors including N. J. Hitchin [23,24], B. Dubrovin [16], B. Dubrovin -M. Mazzocco [18], P. Boalch [6,7,8,9,10], A. V. Kitaev [37,38], A. V. Kitaev -R. Vidūnas [39,61], K. Iwasaki [26]. The classification of algebraic solutions to the Painlevé VI equation was achieved by Lisovyy and Tykhyy [42]. We give some examples of potential vector fields corresponding to algebraic solutions to the Painlevé VI equation. Some other examples and related topics are found in [29,30,31]. In Appendix A, we give a proof of that an Okubo system in several variables is equivalent to an isomonodromic deformation of an Okubo system. Especially, we show that there is a correspondence between Okubo systems in several variables of rank three and generic solutions to the Painlevé VI equation with the help of the result of Jimbo and Miwa [28] (Proposition A.1). In Appendix B, we explain a method of constructing an Okubo system in several variables of rank n from a completely integrable Pfaffian system of rank (n − 1). This construction is used in We close this introduction with addressing some problems. The first one is to construct algebraic solutions to higher order Painlevé equations. As stated above, a solution to the extended WDVV equation yields an isomonodromic deformation of an Okubo system. Recently higher order Painlevé equations are obtained by several authors (e.g. [36,59]) based on the classification theory on linear differential equations due to Oshima. Starting from polynomial potential vector fields corresponding to (real and complex) reflection groups of higher rank, it may be expected that one obtain algebraic solutions to those higher order Painlevé equations. Those solutions may be parametrized by higher-dimensional algebraic varieties. The second one is related with the theory of unfolding of isolated hypersurface singularities. Originally, the view of K. Saito is that flat coordinates arise from versal deformations of isolated hypersurface singularities. Let us focus our attention to the case of ADE singularities. One can define flat coordinates of ADE type from versal deformation of ADE singularities. Then the following question naturally arises: Can we relate flat coordinates for complex reflection groups with "non-versal families" of hypersurface singularities? The third one is related with potential vector fields having polynomial entries. Classification of polynomial prepotentials were conjectured by Dubrovin [16] and proved by Hertling [21]: Any polynomial prepotential comes from the Frobenius manifold structure on the orbit space of a real reflection group. Surprisingly enough, there are some examples in Section 7 whose potential vector fields have polynomial entries but the corresponding flat structures are not isomorphic to one on the orbit space of any complex reflection group. To classify polynomial potential vector fields would be an interesting problem.

Extension of Okubo systems to several variables case
In this section, we start with introducing a special type of a system of ordinary linear differential equations which is called an Okubo system. Then we introduce an extension of an Okubo system to several variables case as a completely integrable Pfaffian system which is called an Okubo system in several variables (Definition 2.3). An Okubo system in several variables is equivalent to an isomonodromic deformation of an Okubo system as we will show in Appendix A (cf. Remark 2.2) and it turns out to be a universal integrable deformation of an Okubo system (Remark 2.3).
Let T and B ∞ be n × n-matrices. If T is a diagonalizable matrix, the system of ordinary linear differential equations is called a system of differential equations of Okubo type or shortly an Okubo system ( [45]). The aim in this section is to extend (4) to a completely integrable Pfaffian system of several variables in the form where B (z) and B (i) are n × n matrices whose entries depend on (z, x). As usual, x = (x 1 , . . . , x n ) denotes a coordinate of C n . We assume the existence of n × n matrices T = T (x) and B ∞ so that B (z) = −(zI n − T ) −1 B ∞ and satisfy the conditions: Let H(x, z) = det(zI n − T (x)) be the characteristic polynomial of T (x), which is a monic polynomial in z of degree n and analytic in x: H(x, z) = z n −S 1 (x)z n−1 +· · ·+(−1) n S n (x). We assume the following condition on H(x, z): ..,n = 0 at generic points of U.
Taking a smaller domain W ⊂ U \ ∆ H appropriately, the eigenvalues z 1 (x), . . . , z n (x) of T can be considered single-valued holomorphic functions in x on W by fixing their branches and we can take an invertible matrix P = P (x) whose entries are single-valued holomorphic functions on W such that We also assume where O U denotes the ring of holomorphic functions on U.
Decompose B (z) into partial fractions on W where B (z) i (i = 1, . . . , n) are independent of z. In addition to (A1)-(A4), we assume the condition: and that B (i) ǫ = ∂E ∂x i E −1 in terms of a matrix E whose entries are holomorphic functions on U. If λ i = λ j for any i = j, the matrix E turns out to be a diagonal matrix E = diag[ǫ 1 , . . . , ǫ n ]. Moreover, it holds that on W . In particular, T,B (i) (i = 1, . . . , n) are mutually commutative: Proof. In virtue of the definition of B (z) , we can write where In virtue of the assumption (A4), we can write As the first step, we are going to show m i ≤ n. From the integrability condition of (5), we have The equation (14) combined with (12) and (13) implies that the left hand side of the resulting equation is a polynomial in z of degree m i + n − 1 and its right hand side is of degree 2n − 2. Besides, the coefficient of the term z m i +n−1 in the left hand side reads As a consequence we find that m i ≤ n, because (HB (z) ) n−1 = −B ∞ is diagonal and is decomposed into partial fractions as follows on W : where B (i) ǫ = (HB (i) ) n is a matrix which is defined on U. If λ i = λ j for any i = j, we see that (HB (i) ) n is a diagonal matrix since [(HB (i) ) n , (HB (z) ) n−1 ] = O. Paying attention to the value at z = ∞ of the equation obtained by substituting (15) for the integrability condition of the Pfaffian system (5), we have

It means that B
(i) ǫ is written by a matrix E as B As the second step, we shall show the equalities for the residue matrices B (7) and (15). We have by substituting (7) and (15) for the integrability condition. Now fix j ∈ {1, . . . , n}. Since rank (B From (17) and (18), we have j , which and (17) again, imply the equalities (16).
As the last step, we take a matrix P that satisfies (9). Then and from (16), we have Hence we obtain This proves (9), (10) and thus (11) on W . Since B ∞ is invertible, we find thatB (i) ∈ O n×n U . In virtue of the identity theorem, (11) holds on U. We have thus proved the lemma completely.
In virtue of Lemma 2.1, we restrict ourselves to the case B (i) ǫ = O without loss of generality by applying a gauge transformation to Y . Namely we consider the Pfaffian system with then the Pfaffian system (19) is completely integrable. Moreover, in case where λ i = 0 (i = 1, . . . , n), the system of equations (20), (21), (22) is necessary and sufficient condition for that (19) is completely integrable.
Proof. The lemma is clear from the relations Definition 2.3. The completely integrable Pfaffian system (19) is called a system of differential equations of Okubo type in several variables or shortly an Okubo system in several variables.
Remark 2.2. The completely integrable Pfaffian system (19) can be regarded as a Lax formalism of the isomonodromic deformation of an Okubo system (4). Indeed (19) is equivalent to Then the system of nonlinear differential equations (21) is equivalent to the Schlesinger system dB (See Appendix A for details.) Remark 2.3. By (10) and the assumption (A3), we see that an Okubo system in several variables is a universal integrable deformation of an Okubo system (4), which is uniquely determined by (4) up to changes of independent variables. (The uniqueness follows also from that of the isomonodromic deformation (23), (24) which is proved in [27].) Lemma 2.4. The variables x 1 , . . . , x n can be taken so that T = T (x) satisfies the following condition: In the following, we always assume Condition (T).

Logarithmic vector fields
In this section, we study a divisor defined by det(−T (x))) which is the singular locus of an Okubo system in several variables. A main purpose of this section is to prove that the vector fields defined by (34) forms a unique standard system of generators of logarithmic vector fields along the divisor when the divisor is free (Lemma 3.9, cf. Remark 3.2).

Logarithmic vector fields
We employ the notations x ′ = (x 1 , . . . , x n−1 ) and We assume that the domain U ⊂ C n (and W ⊂ U \ ∆ H resp.) has the form of U = U ′ × C ⊂ C n−1 × C (and W = W ′ × C ⊂ U ′ × C resp.). Note that h(x) is a monic polynomial in x n of degree n: Here we note that the assumption (A3) for H(x, z) = h(x ′ , x n + z) is equivalent to We give the definitions of logarithmic vector field along D and free divisor following K. Saito [52]: Let Der(− log D) be the set of logarithmic vector fields along D, which is naturally an We rewrite the assumption (A3) in terms of meromorphic logarithmic vector fields (which will be used in the proof of Lemma 6.1): be any meromorphic logarithmic vector field along D (whose coefficients do not depend on x n ). Then V = 0.
Noting (26), the equation (29) has a non-zero solution if and only if (A3) for H(x, z) does not hold. This proves the lemma. and and We define the vector fields V The two matrices M 0 and M 1 are determined by . . , z 0 n ] and M 1 F = P h using elementary elimination methods. By the construction, M 0 and δ h M 1 are holomor- we find that the entries of (30), (31), (34).
Then the equality (33) implies that M V (h) is a Saito matrix and D is free.

Global case
In this subsection we assume that the function h(x ′ , x n ) is a polynomial in (x ′ , x n ) and weighted homogeneous with respect to a weight w(·) with In this case, we replace O U ′ and M U ′ in the previous subsection by C[x ′ ] and C(x ′ ) respectively, and assume that the integrable system (25) is weighted homogeneous, that is, all the entries of B is the Euler operator.
Proof. It is clear from that the Euler operator is a logarithmic vector field along D and the proof of Lemma 3.4. Since between {w(x i )} and {w(V (h) i )}. As for the reduced form (25), we give a lemma.
(ii) There is a weighted homogeneous matrix C(x) such that Proof. Proof of (i). From (9) and (10), we have Proof of (ii). From (22), be the weighted homogeneous part ofC ij (x) of the weight w(T ij ). Then the matrix C(x) = (C ij (x)) i,j=1,...,n satisfies the first equality of (39). The second equality of (39) then coincides with the first one of (38).
. . , λ n ] as before. Then, from (38), we have From the equality ∂ Since B (n) Lemma 3.7. Let V be a non-zero weighted homogeneous logarithmic vector field along D. Then w(V ) ≥ 0.

For rational logarithmic vector fields
n for a non-zero constant number c. Then the following three facts hold.
i (x), 1 ≤ i ≤ n are also logarithmic vector fields, and hence D is free. ( Since all the entries of R −1 M V (x) are linear function of x n with the coefficients being weighted homogeneous polynomials in x ′ , we have This proves (ii).
..,n forms a unique system of generators of logarithmic vector fields along D such that its Saito Since each homogeneous part of V i are logarithmic vector field with a non-negative weight (Lemma 3.7), all entries of ( Remark 3.2. In the case where h(x) is the discriminant of a well-generated complex reflection group, D. Bessis [5] showed the existence of a system of generators of logarithmic vector fields along D whose Saito matrix M V has the form M V − x n I n ∈ C[x ′ ] n×n . Such a system of generators is called flat in [5] (cf. Section 6).

Saito structure (without metric)
In this section, we review Saito structure (without metric) introduced by Sabbah [51]. (In the sequel, we abbreviate Saito structure (without metric) to Saito structure for brevity.) Proofs for many of statements in this section can be found in the literature [51,40]. [51]). Let X be a complex analytic manifold of dimension n (in this paper, we treat the case where X is a domain U in C n ), T X be its tangent bundle, and Θ X be the sheaf of holomorphic sections of T X. A Saito structure (without a metric) on X is a data consisting of ( △ , Φ, e, E) in (i),(ii),(iii) that are subject to the conditions (a), (b): △ is a flat torsion-free connection on T X, (ii) Φ is a symmetric Higgs field on T X, (iii) e and E are global sections (vector fields) of T X, respectively called unit field and Euler field.
(a) A meromorphic connection ∇ on the bundle π * T X on P 1 × X defined by is integrable, where π is the projection π : P 1 × X → X and z is a non-homogeneous coordinate of P 1 , △ (e) = 0) and satisfies Φ e = Id, where we regard Φ as an End O X (Θ X )-valued 1-form and Φ e ∈ End O X (Θ X ) denotes the contraction of the vector field e and the 1-form Φ.
Remark 4.1. To the Higgs field Φ there associates a product ⋆ on Θ X defined by ξ ⋆ η = Φ ξ (η) for ξ, η ∈ Θ X . The Higgs field Φ is said to be symmetric if the product ⋆ is commutative. The condition Φ e = Id in Definition 4.1 (b) implies that the field e is the unit of the product ⋆. The integrability of the connection ∇ implies that of the Higgs field Φ, which is equivalent to the associativity of ⋆. So the product ⋆ associated to a Saito structure is commutative and associative.
Since the connection △ is flat and torsion free, we can take a flat coordinate system (t 1 , . . . , t n ) such that △ (∂ t i ) = 0 (i = 1, . . . , n) at least on a simply-connected open set of X. We assume the existence of a flat coordinate system (t 1 , . . . , t n ) on X replacing X by such an open set if necessary. In addition, we assume the following conditions: In this paper, a function f ∈ O X is said to be weighted homogeneous with a weight w(f ) ∈ C if f is an eigenfunction of the Euler operator: We fix the basis {∂ t 1 , . . . , ∂ tn } of Θ X and introduce the following matrices: (ii) T and B ∞ are the representation matrices of −Φ(E) and △ E respectively, namely We assume that −Φ(E) is generically regular semisimple on X, that is the discriminant of det(z − T ) does not identically vanish on X.
Proof. It is straightforward.   (46) is nothing but the integrability condition of the Pfaffian system In other words, the existence of a Saito structure yields an Okubo system in several variables (47). In the next section, we will find a criterion for that, for an Okubo system in several variables, there exists a Saito structure which yields the given Okubo system in several variables.  (43) is written in the following matrix form with respect to the flat coordinate system: The system of equations (46) is equivalent to the integrability condition of (48). The system of ordinary linear differential equations has an irregular singularity of Poincaré rank one at z = 0 and a regular singularity at z = ∞, which is called a Birkhoff normal form. So (48) may be regarded as a universal integral deformation of a Birkhoff normal form. A Birkhoff normal form can be transformed into an Okubo system using a Fourier-Laplace transformation.
There is a unique matrix C such that Proof. This lemma can be proved in a way similar to Lemma 3.6.
Proof. As in Lemma 4.4, define vector fields Then Lemma 4.4 shows that each V i is logarithmic along D. Then similarly to Lemma 3.5, it holds that V 1 = E, from which we have −T nj = w j t j . It is equivalent to C nj = t j by Lemma 4.5.
Proposition 4.7 (Konishi-Minabe [40]). There is a unique n-tuple of analytic functions and that g j is weighted homogeneous with w(g j ) = 1 + w j . The vector g (or precisely the vector field G = n i=1 g i ∂ t i ) is called a potential vector field.
Proof. By the symmetry of the Higgs field Φ, it holds that Then g j is uniquely given by Proposition 4.8. The potential vector field g = (g 1 , . . . , g n ) is a solution to the following system of nonlinear equations: Proof. The associativity of ⋆ (i.e., the integrability of Φ) is equivalent toB (k)B(l) = B (l)B(k) . We obtain (50) if we rewrite this condition in terms ofB (51) follows fromB (n) = I n (i.e., Φ e = Id).
Remark 4.4. The notion of "F -manifolds with compatible flat structures" was introduced by Manin [44] as a generalization of Frobenius manifolds. This notion does not require the existence of an Euler field. "Potential vector field" in Proposition 4.7 is called "local vector potential" in Manin's framework [44]. And the associativity conditions (50), (51) are called "oriented associativity equations" in [44]. The authors were informed these facts by A. Arsie and P. Lorenzoni.
Conversely, starting with a solution of (50), (51) and (52), it is possible to reconstruct a Saito structure. Proposition 4.10. Take constants w j ∈ C, j = 1, . . . , n satisfying w i − w j ∈ Z and w n = 1 and assume that g = (g 1 , . . . , g n ) is a holomorphic solution of (50), (51) and (52) on a simply connected domain U in C n . Then there is a Saito structure on U which has (t 1 , . . . , t n ) as a flat coordinate system. In addition, the Saito structure is semisimple (i.e.
Let J be an n × n-matrix with J ij = δ i+j,n+1 , i, j = 1, . . . , n, where δ ij denotes Kronecker's delta, and, for an n × n matrix A, define A * by A * = JA t J.
Proposition 4.11. Given a Saito structure on X, the following conditions are mutually equivalent: (i) For appropriate normalization of the flat coordinate system, it holds that C * = C.
Here, as stated in Remark 4.5, a flat coordinate system admits indetermination of multiplication by constants. "Normalization" in the above conditions means to fix this indetermination.
Proof. (i) ⇔ (ii) By definition, C * = C is equivalent to that CJ is a symmetric matrix.
we find that the symmetry of CJ is equivalent to the existence of F ∈ O X such that . . , n for some r ∈ C. Note that, if C * = C holds, then it also holds that T * = T andB (i) * =B (i) , i = 1, . . . , n. By the integrability condition, we have Taking the difference of these two equalities, we have Since T is written as Besides, we note that T nj = −w j t j = 0, j = 1, . . . , n. Then it follows that B ∞ + B * ∞ = diag[w 1 + w n , w 2 + w n−1 , . . . , w n + w 1 ] is a scalar matrix. Next, we show the existence of a metric with the desired properties. Under the assumption that CJ is a symmetric matrix, define a metric η by η(∂ t i , ∂ t j ) = J ij , i, j = 1, . . . , n for the flat coordinate system (t 1 , . . . , t n ). Then, on one hand, we have on the other hand which concludes the compatibility of η to the product. The horizontality and the homogeneity hold obviously. (iii) ⇒ (i) By the horizontality, it follows that η(∂ t i , ∂ t j ) (i, j = 1, . . . , n) are constants. From the homogeneity, it holds that which implies that we can take a flat coordinate system (t 1 , . . . , t n ) such that η(∂ t i , ∂ t j ) = J ij and we do so in the sequel. Then it holds that which impliesB (k) * =B (k) , k = 1, . . . , n. Hence T * = T and C * = C also hold.
The function F in Proposition 4.11 is called a prepotential in [16] and a potential in [51]. It is well known that the prepotential satisfies the WDVV equation (cf. [16]).
5 Flat structure on the space of isomonodromic deformations We start with an Okubo system in several variables: Let the assumptions on (58) be same as in Section 2. The purpose of this section is to find a necessary and sufficient condition for that the Okubo system in several variables (58) arises from a Saito structure. Here, to avoid the confusion, we state the precise definition of that an Okubo system in several variables (58) arises from a Saito structure.
Definition 5.1. We say that an Okubo system in several variables (58)  We consider (58) restricted on an appropriate small domain W ⊂ U \ ∆ H so that we can take an invertible matrix P = P (x) such that Then the homomorphism ϕ from T W to C n (W ) defined by ϕ(∂ x i ) = F (i) (e n ) is isomorphic if and only if n j=1 P nj = 0, which can be seen from e n = (P n1 , . . . , P nn )P −1 (e 1 , . . . , e n ) t and thus ϕ(∂ x i ) = − ∂z 1 ∂x i P n1 , . . . , − ∂z n ∂x i P nn P −1 (e 1 , . . . , e n ) t .
Put e := ϕ −1 (e n ), E := ϕ −1 (F (e n )) and introduce a Higgs field Φ on T W by Besides, define a connection △ on T W by △ Lemma 5.3. We consider an Okubo system in several variables (58). The following two conditions are equivalent to each other: Proof. From (21), it holds that from which and (10) we have Hence we obtain that at any point on U. If (58) arises from a Saito structure on U, the set of variables t j := −(λ j − λ n + 1) −1 T nj , j = 1, . . . , n provides a flat coordinate system.
Proof. First, we assume that (58) arises from a Saito structure. Take a flat coordinate system (t 1 , . . . , t n ) as independent variables of (58). Then it holds that T = T and T nj = −w j t j , from which we have . . . ∂Tnn ∂tn = (−1) n w 1 · · · w n = 0.
Conversely, we assume (59). In virtue of Lemmas 5.2 and 5.3, (58) arises from a Saito structure on W ⊂ U \ ∆ H . We also see that {t j = −(λ j − λ n + 1) −1 T nj } is a flat coordinate system. Then E := n k=1 (λ k − λ n + 1)t k ∂ t k , e := ∂ tn and Φ := n j=1B (j) dx j = j,k ∂x j ∂t kB (j) dt k satisfy the axiom of Saito structure on W . Due to the identity theorem, they satisfy it also on U. Hence (58) arises from a Saito structure on U.
Theorem 5.4 is used to construct flat coordinates in Sections 6 and 7.
Remark 5.1. Uniqueness of the Saito structure corresponding to an Okubo system in several variables follows from the following argument. An Okubo system in several variables (58) admits the following types of gauge freedom: One is similarity transformations by a constant matrix C such that CB ∞ C −1 = B ∞ , which corresponds to the freedom of flat coordinate systems mentioned in Remark 4.5. The other is permutations on matrix entries. Let σ be a permutation on the set {1, 2, . . . , n}. For an Okubo system in several variables (58), we consider the following change of dependent variables: Then Y σ satisfies a new Okubo system in several variables where T σ denotes the matrix whose (i, j)-entry is defined by T σ ij := T σ(i),σ(j) and the similar holds forB (i)σ , B σ ∞ . If ∂T σ(n),1 ∂x 1
∂T σ(n),1 ∂xn · · · ∂T σ(n),n ∂xn = 0 holds at any point on U, the space of independent variables of (58) can be equipped with a Saito structure by applying Theorem 5.4 to the new system (60), which differs in general from that obtained from the original system (58). Therefore the space of independent variables of (58) can be equipped with at most n mutually different Saito structures up to isomorphisms. (Note that, if σ fixes n (i.e. σ(n) = n), then the Saito structure obtained from (60) is isomorphic to that obtained from (58). Indeed, σ induces only the permutation on the flat coordinates (t 1 , . . . , t n ) → (t σ 1 := t σ(1) , . . . , t σ n := t σ(n) = t n ).) We see from this argument that, if one of eigenvectors of B ∞ is designated, the corresponding Saito structure is uniquely specified up to isomorphisms (provided that the condition in Theorem 5.4 is satisfied). For example, if all of the eigenvalues are real numbers and satisfy λ 1 ≤ · · · ≤ λ n−1 < λ n , then we can distinguish λ n (and the eigenvector belonging to it) from the remaining eigenvectors and a unique Saito structure can be determined corresponding to it. (The flat structure on the orbit space of a well-generated unitary reflection group introduced in Section 6 is in this case.) The initial value problem for regular Saito structures (including the non-semisimple case) is treated in [35].
Corollary 5.5. We consider the case of n = 3. There is a correspondence between solutions satisfying the semisimplicity condition (SS) in Proposition 4.10 to the extended WDVV equation , i, j, k, l = 1, 2, 3, w k t k ∂g j ∂t k = (1 + w j )g j , j = 1, 2, 3.
and generic solutions to the Painlevé VI equation Let T be the 3 × 3 matrix whose entries are given by and take P such that P −1 T P is a diagonal matrix. Let and θ ∞ = w 1 −w 2 , where we remark that r i is an explicit form of that in (A5) in Section 2. Then the correspondence between the parameters is given by Proof. Since the condition (59) [43] shows that the three-dimensional regular semisimple bi-flat F -manifolds are parameterized by generic solutions to the (fullparameter) Painlevé VI equation. Recently it is proved in [3,41] that bi-flat F -manifold is an equivalent notion to Saito structure. Therefore Corollary 5.5 provides another proof of Arsie-Lorenzoni's result. The proof here makes clear the relationship between flat structures and isomonodromic deformations.
In [2], Arsie and Lorenzoni study the relationship between three-dimensional regular non-semisimple bi-flat F -manifolds and Painlevé IV and V equations. Then it is naturally expected that there is a correspondence between solutions to the extended WDVV equation not satisfying the semisimple condition (SS) and the isomonodromic deformations of generalized Okubo systems introduced in [34]. This is treated in [35].
Theorem 5.4 on the existence of a Saito structure associated with a given Okubo system in several variables can be rephrased as follows.
Theorem 5.6. We consider the reduced form (25) in Remark 2.5 of an Okubo system in several variables (58) with det B ∞ = 0: Then the following two conditions are equivalent: (i) It holds that Proof. From (61), we see that (∂y n /∂x 1 , . . . , ∂y n /∂x n ) t = MY 0 holds, where we put  Proposition 5.7. Assume that the reduced form of an Okubo system in several variables (61) arises from a Saito structure on U and take the flat coordinate system (t 1 , . . . , t n ) = (C n1 , . . . , C nn ) as the independent variables of (61). Then there is a function y = y(t) such that Y 0 = (∂y/∂t 1 , . . . , ∂y/∂t n ) t .
Proof. We consider the "contiguous" equation of (61): For any solution Y Hence we can take y = y (−) n as the desired function.

Flat generator system for invariant polynomials of a complex reflection group
In this section, we treat a problem on the existence of flat basic invariants for a complex reflection group. In the case of real reflection groups, K. Saito [53] proved the existence of flat basic invariants (see also [54]). For a well-generated complex reflection group G, we construct a Saito structure on the orbit space of G using an Okubo system in several variables called G-quotient system (Theorem 6.2). (The G-quotient system is a Pfaffian system whose fundamental system of solutions consists of derivatives by logarithmic vector fields of linear coordinates on the standard representation space of G and its monodromy group is isomorphic to G, see Theorem 6.2 and Remark 6.1.) As a consequence, we find that the potential vector field of the Saito structure for a well-generated complex reflection group G has polynomial entries (Corollary 6.8). It is underlined that the following proof of Theorem 6.2 is constructive i.e. it contains an algorithm which provides explicit computation of the flat generator system of G-invariants and the potential vector field for each well-generated complex reflection group G. See [29] for explicit formulas of potential vector fields for exceptional groups. (There is a procedure to construct flat basic invariants directly from a potential vector field corresponding to a finite complex reflection group, see Remark 6.2.) Let G be a finite irreducible complex (unitary) reflection group acting on the standard representation space U n = {(u 1 , u 2 , . . . , u n ) | u j ∈ C}, and let F i (u) of degree d i , 1 ≤ i ≤ n be a fundamental system of G-invariant homogeneous polynomials. We assume that We define a coordinate functions on X := U n /G by, Let D ⊂ X be the branch locus of π G : U n → X. Let h(x) be the (reduced) defining function of D in the coordinates x = (x 1 , x 2 , . . . , x n ). We assume that G is well generated (see e.g. [5]). Then it is known that h(x) is a monic polynomial in x n of degree n ( [5]). We define a weight w(·) by w(x i ) = d i /d n . Then h(x) is a weighted homogeneous polynomial in x. It is known that D is free ( [47,60]). Here we give a key lemma for the discriminant h(x). Proof. It is known ( [5]) that there is a generator {V 1 , . . . V n } of logarithmic vector fields along D such that V i = n j=1 v ij (x) ∂ x j , 1 ≤ i ≤ n are weighted homogeneous and satisfy In particular, it holds that Let I = {i | a i (x) = 0}, and assume I = ∅. Let i 0 ∈ I be such that which contradicts the equality (64) with j = n + 1 − i 0 . This implies that I = ∅, that is, V = 0. Then Lemma 3.2 asserts that H(x, z) satisfies (A3). Let In this section, we prove the following theorem:   (34) and (31) with respect to h(t). Let C(t) be the (weighted homogeneous) matrix satisfying V Then, for any homogeneous linear function y(u) of u, satisfies the Okubo system Note that the n-th entry of Y ′ equals y(u), that is d n V (h) 1 (t) y(u) = y(u). (ii) It holds that C n,j (t) = t j , 1 ≤ j ≤ n, and hence {t j } gives a flat coordinate system on X associated to (67) by Theorem 5.4.
If d 1 < d 2 · · · < d n , then F f l i (u) are unique up to constant multiplications. Definition 6.3. We call {F f l i (u)} a flat generator system of G-invariant polynomials or flat basic invariants of G.
Remark 6.1. A generating system in [20] is a differential equation with a single unknown satisfied by homogeneous linear functions y(u). In [32], the generating system is rewritten as a Pfaffian system and called a G-quotient system. The equation (67) is a G-quotient system in a form of Okubo type.
Again, we let x i = F i (u), 1 ≤ i ≤ n for arbitrarily given fundamental system of G-invariant homogeneous polynomials, and h(x) be the defining function of D in the coordinates x.
for any homogeneous linear function y(u) of u. Then we haveŷ n = w(y)y = (1/d n ) y, and Lemma 6.6. All entries of h(x)B (k) (x) are weighted homogeneous polynomials in x, and Y satisfies the system of differential equations and since This proves the lemma.
Lemma 6.7. LetŶ ,B (k) be the same as in Lemma 6.6. Then there is an upper triangular matrix R(x ′ ) ∈ GL(n, C[x ′ ]) such that, if we put then the system of differential equations satisfied by Y 0 is an Okubo system in several variables with (B can be chosen such that all the entries are weighted homogeneous with w(R(x ′ ) i,j ) = w(ŷ i ) − w(ŷ j ), and R(0) = I n .
In particular, it holds that y n =ŷ n = (1/d n )y(u).
In particular, we have which shows R i are diagonal, andB ∞ (x ′ ) is an upper triangular matrix with the diagonal elements −w(ŷ i ). Then, by elementary linear algebra, we find that there is an upper By construction of R(x ′ ), we find that all entries R(x ′ ) i,j are weighted homogeneous with Since the residue matrix ofB (n) dx n at zeros of h(x ′ , x n ) is of rank one and diagonalizable, so is the residue matrix of B (n) 0 dx n . Consequently, we find that there exists a n × n matrix at least at any generic point x ′ by an argument similar to the one in Appendix B. Let Then, by induction, we find In particular, it holds that T 0 (x ′ ) = s 1 (x ′ )I n − C n−2 (x ′ )B −1 ∞ , which implies that all entries of T 0 (x ′ ) are weighted homogeneous polynomials in x ′ .
Finally we prove that the differential system (76) satisfies (B we have m k ≤ n for k ≤ n − 1 and m n = n − 1. For m k = n, from the equality [(hB , Then (21) impliesB Now we prove Theorem 6.2.
Proof of Theorem 6.2. From now on, to avoid confusion, we denote by h x (x) the defining function of D in the coordinates x.
In particular V ′ 1 = d n V 1 . Then From Theorem 5.6, it holds that ∂xn · · · ∂T ′ nn ∂xn = 0, which implies that t j = C ′ nj (x), 1 ≤ j ≤ n form a coordinate system on X. Note , the equalities t j = C ′ nj (x), 1 ≤ j ≤ n are solved by weighted homogeneous polynomials x j = x j (t), 1 ≤ j ≤ n, and hence {F f l j (u)} is a fundamental system of G-invariant polynomials. Let Then h t (t) is the defining function of D in the coordinates t; C nj (t) = t j , 1 ≤ j ≤ n; The and Lemma 4.4 implies that n j=1 (−T (t) ij ) ∂ t j , 1 ≤ i ≤ n are logarithmic vector fields along D. From these two properties and Lemma 3.9, we find M V (h t ) (t) = −T (t).
From the "(i) ⇒ (ii)" part of the proof of Theorem 5.6, (82) implies which is equivalent to (66). This completes the proof of the existence of a flat generator system of G-invariant polynomials satisfying (i) and (ii) in Theorem 6.2. The uniqueness of a flat generator system under the condition d 1 < d 2 < · · · < d n is clear from Remarks 5.1 and 4.5.
Corollary 6.8. The potential vector field g = (g 1 , . . . , g n ) corresponding to the G-quotient system (67) is a polynomial in t 1 , . . . , t n . In other words, the potential vector field of the G-quotient system yields a polynomial solution to the extended WDVV equation.
Proof. By the above construction, h(t) and T = −M V (h) (t) are polynomials in t 1 , . . . , t n . Thus, C(t) and g are also polynomials. In fact, g j are given by Remark 6.2. Irreducible finite complex reflection groups are classified by Shephard-Todd [58]: Except for rank 1 groups, there are two infinite families A n , G(pq, p, n), plus 34 exceptional groups G 4 , G 5 , . . . , G 37 . Explicit forms of potential vector fields for the exceptional groups are discussed in [29]. It is possible to directly compute the explicit form of flat basic invariants of a complex reflection group G from its potential vector field by the following procedure: Let (x 1 , . . . , x n ) = (F 1 (u), . . . , F n (u)) be arbitrary basic invariants of G and write down the discriminant h x (x) of G in terms of x. On the other hand, one can write down the discriminant h t (t) of G in terms of t from the potential vector field in the manner described in Section 4 (i.e. h t (t) := det(−T ), where T is constructed from the potential vector field as in the proof of Proposition 4.10). Find a weight preserving coordinate change t = t(x) such that h t (t(x)) = h x (x). Then F f l 1 (u), . . . , F f l n (u) := t 1 (F 1 (u), . . . , F n (u)), . . . , t n (F 1 (u), . . . , F n (u)) provides flat basic invariants of G.
Remark 6.3. The papers [46,17] treat Frobenius structures constructed on the orbit spaces of Shephard groups (which consist a subclass of complex reflection groups). In the case of some Shephard groups, one can find that the Saito structure constructed in Theorem 6.2 which corresponding to the G-quotient system of the Shephard group does not have a prepotential. Therefore in this case, the Saito structure in Theorem 6.2 is distinct from the Frobenius structure treated in [46,17]. This phenomenon is naturally explained in the framework of this article: It is known that to each Shephard group there is an associated Coxeter group whose discriminant is isomorphic to that of the Shephard group. Then we see that there are (at least) two Okubo systems in several variables on the orbit space of the Shephard group which have singularities along the discriminant: one is the G-quotient system of the Shephard group, the other is the G-quotient system of the associated Coxeter group. The Frobenius structure on the orbit space of a Shephard group described in [46,17] corresponds to the G-quotient system of the associated Coxeter group in this picture.

Examples of potential vector fields corresponding to algebraic solutions to the Painlevé VI equation
In this section we show some examples of potential vector fields in three variables which correspond to algebraic solutions to the Painlevé VI equation.
Algebraic solutions to the Painlevé VI equation were studied and constructed by many authors including N. J. Hitchin [23,24], B. Dubrovin [16], B. Dubrovin -M. Mazzocco [18], P. Boalch [6,7,8,9,10], A. V. Kitaev [37,38], A. V. Kitaev -R. Vidūnas [39,61], K. Iwasaki [26]. The classification of algebraic solutions to the Painlevé VI equation was achieved by Lisovyy and Tykhyy [42]. We remark that all the algebraic solutions in the list of [42] had previously appeared in the literature (see [11] and references therein). One of the principal aims of our study is the determination of a flat coordinate system and a potential vector field for each of such algebraic solutions. In spite that this aim is still not succeeded because of complexity of computation, we show some examples of potential vector fields. Some of the results below are already given in [29]. Other examples can be found in [31]. From the construction of polynomial potential vector fields corresponding to finite complex reflection groups of rank three (Corollary 6.8) and Corollary 5.5, we obtain a class of algebraic solutions to the Painlevé VI equation. The relationship between finite complex reflection group of rank three and solutions to the Painlevé VI equation was first studied by Boalch [6]. (More precisely speaking, it was conjectured in [6] that the solutions obtained from finite complex reflection groups by his construction are algebraic and this conjecture was proved in his succeeding papers.) The construction in this paper answers the question 3) in the last part of [6]: "Is there a geometrical or physical interpretation of these solutions?" It is remarkable that there are examples in Sections 7.4,7.5,7.6 whose potential vector fields have polynomial entries but the corresponding flat structures are not isomorphic to one on the orbit space of any finite complex reflection group because the free divisors defined by F B 6 , F H 2 , F E 14 in Sections 7.4,7.5,7.6 respectively are not isomorphic to the discriminant of any finite complex reflection group. The existence of these examples suggests that an analogue of Hertling's theorem ( [21]) does not hold in the case of non Frobenius manifolds. It would be an interesting problem to classify all the polynomial potential vector fields.
To avoid the confusion, we prepare the convention which will be used in the following. We treat the case n = 3. Let t = (t 1 , t 2 , t 3 ) be a flat coordinate system and g = (g 1 , g 2 , g 3 ) denotes a potential vector field. Let w(t i ) be the weight of t i and assume 0 < w(t 1 ) < w(t 2 ) < w(t 3 ) = 1. The matrix C is defined by In this section, an algebraic solution LTn means "Solution n" in Lisovyy-Tykhyy [42], pp.156-162.

Algebraic solutions related with icosahedron
We treat the three algebraic solutions to Painlevé VI obtained by Dubrovin [16] and Dubrovin-Mazzocco [18].

Icosahedral solution (H 3 )
In this case, and there is a prepotential defined by Then it follows from the definition that g j = ∂ t j F (j = 1, 2, 3) give the potential vector field g = (g 1 , g 2 , g 3 ).
We don't enter the details on this case. See [16,18].
Great icosahedral solution (H 3 ) ′ Let (t 1 , t 2 , t 3 ) be a flat coordinate system and their weights are given by We introduce an algebraic function z of t 1 , t 2 defined by the relation It is clear from the definition that w(z) = 1 5 . In this case, we consider the algebraic function of (t 1 , t 2 , t 3 ) defined by Then we see that F is a solution to the WDVV equation. Indeed, we first define Then ∂ t i C (i = 1, 2, 3) commute to each other. This condition is equivalent to that F is a solution to the WDVV equation.
Great dodecahedron solution (H 3 ) ′′ Let (t 1 , t 2 , t 3 ) be a flat coordinate system and their weights are given by We introduce an algebraic function z of t 1 , t 2 defined by the relation It is clear from the definition that w(z) = 1 3 . In this case, we consider the algebraic function of (t 1 , t 2 , t 3 ) defined by Then F is also a solution of the WDVV equation. The following two cases are related with the complex reflection group ST27. Solution 38 of P. Boalch [8] (LT26) In this case, is regarded as the discriminant of the complex reflection group ST27, if t 1 , t 2 , t 3 are taken as basic invariants.

Algebraic solutions related with the polynomial F B 6 in [56]
We recall the polynomial which is a defining equation of a free divisor in C 3 (cf. [56]). There are two algebraic solutions which are related with the polynomial F B 6 .

Algebraic solutions related with the polynomial F H 2 in [56]
We recall the polynomial F H 2 = 100x 3 y 4 + y 5 + 40x 4 y 2 z − 10xy 3 z + 4x 5 z 2 − 15x 2 yz 2 + z 3 which is a defining equation of a free divisor in C 3 (cf. [56]). There are two algebraic solutions which are related with the polynomial F H 2 .

A Okubo systems in several variables and isomonodromic deformations
Let us recall that an Okubo system in several variables of rank n is defined as follows: Especially in the case of n = 3, this equivalence can be related to the Painlevé VI equation (a proof of the following Proposition A.1 is given at the end of this Appendix A): Proposition A.1. We consider (85) for n = 3. Then there is a correspondence between Okubo systems in several variables of rank 3 and generic solutions to the Painlevé VI equation with the parameters (θ 0 , θ 1 , θ t , θ ∞ ) = (r 1 + λ 3 , r 2 + λ 3 , r 3 + λ 3 , λ 1 − λ 2 ), where λ i and r i are defined in (A2) and (A5) respectively in Section 2.
Here, we briefly review the theory of isomonodromic deformation (following to [27,28]). We consider a system of ordinary differential equations of rank m whereŶ ∞ (z −1 ) is a convergent power series of z −1 such thatŶ (∞) (0) = I m . Fixing paths from z = ∞ to z = a i (i = 1, . . . , n), the analytic continuations of Y (∞) to z = a i (i = 1, . . . , n) along the paths are described as follows: whereŶ (i) (z − a i ) is a convergent power series of z − a i such thatŶ (i) (0) = I m , Λ i = G i B i G −1 i and C i is a constant invertible matrix. Now we deform (86) moving a 1 , . . . , a n as variables, namely we suppose that B i (i = 1, . . . , n) are functions of a 1 , . . . , a n . Then Y (∞) is a function of z, a 1 , . . . , a n and r (i) j , G i , C i are functions of a 1 , . . . , a n . A deformation of (86) with variables a 1 , . . . , a n is said to be an isomonodromic deformation (or monodromy preserving deformation) if r (i) j (i = 1, . . . , n, ∞, j = 1, . . . , m) and C i (i = 1, . . . , n) are constants independent of a 1 , . . . , a n . The following fact is well known (see [27], [28] for instance): Fact 1. A deformation of (86) is isomonodromic if B i (i = 1, . . . , n) satisfy the following system of differential equations: which is called the Schlesinger system.
Returning to our situation, we consider an Okubo system in several variables of rank n (85). Let z 1 , . . . , z n be the roots of det(zI n −T ), and decompose B (z) into partial fractions Lemma A.2. The system of equations (20), (21), (22) is equivalent to the Schlesinger system dB Proof. On one hand, from Lemma 2.1, (20), (21), (22) are equivalent to the integrability condition of (85). On the other hand, it is known that the Schlesinger system (89) is equivalent to the integrability condition of the Pfaffian system Therefore, it is sufficient to show that the two Pfaffian systems (85) and (90) are equivalent to each other. Changing the independent variables (x 1 , . . . , x n ) of (85) to (z 1 , . . . , z n ), (85) is rewritten as Let E i be the matrix whose (j, k)-entry are defined by (E i ) jk = δ ij δ ik . Then we have n j=1B (j) ∂x j ∂z i = −P E i P −1 and thus −(zI n − T ) −1 n j=1B (j) ∂x j ∂z i B ∞ = P E i P −1 B ∞ /(z − z i ).
It is straightforward to check that B As a preparation to prove Proposition A.1, we quote from [28, Appendix C] (a concise explanation is found in [7]) the following fact on the relation between the Painlevé VI equation and the isomonodromic deformation of a system of linear differential equations of rank 2. (Notations in Fact 2 are valid only in this part.) Fact 2. We consider a Pfaffian system of rank 2 dZ = A(x, t)dx + B(x, t)dt Z, where and A 0 , A 1 , A t are 2 × 2 matrices whose entries are functions of t (independent of x). Put A ∞ := −A 0 − A 1 − A t . We assume that A 0 , A 1 , A t are rank 1 matrices and that A ∞ is a diagonal matrix. Put θ i := trA i (i = 0, 1, t). Then A i (i = 0, 1, t) can be written as follows: From the assumption that A ∞ is diagonal, we have the following relations: We put κ 1 := −(z 0 + θ 0 + z 1 + θ 1 + z t + θ t ), κ 2 := z 0 + z 1 + z t .
From the first equation of (95), we find that the (1, 2)-entry of A(x, t) is of the form for some linear polynomial p(x) in x. Explicitly, p(x) is expressible as p(x) = k(x − y) with k = (t + 1)uz 0 + tvz 1 + wz t , y = k −1 tuz 0 .
Finally we obtain an Okubo system in several variables by extending (113) to an integrable Pfaffian system following Lemma 2.1.