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Regular flat structure and generalized Okubo system

We study a relationship between regular flat structures and generalized Okubo systems. We show that the space of variables of isomonodromic deformations of a regular generalized Okubo system can be equipped with a flat structure. As its consequence, we introduce flat structures on the spaces of independent variables of generic solutions to (classical) Painlev\'e equations (except for PI). In our framework, the Painlev\'e equations PVI-PII can be treated uniformly as just one system of differential equations called the four-dimensional extended WDVV equation. Then the well-known coalescence cascade of the Painlev\'e equations corresponds to the degeneration scheme of the Jordan normal forms of a square matrix of rank four.


Introduction
At the end of 1970's, K. Saito introduced the notion of flat structure in order to study the structure of the parameter spaces of isolated hypersurface singularities. In particular he defined and constructed flat coordinates on the orbit spaces of finite real reflection groups in [32,33] with his collaborators T. Yano and J. Sekiguchi. After their pioneering work, B. Dubrovin [7] unified both the flat structure formulated by K. Saito and the WDVV equation which arises from the 2D topological field theory as Frobenius manifold structure. In his theory on Frobenius manifolds, Dubrovin studied a relationship between Frobenius manifolds and isomonodromic deformations of linear differential equations of certain type. Especially he showed that there is a correspondence between three dimensional regular semisimple Frobenius manifolds (massive Frobenius manifolds) and generic solutions to a one-parameter family of the sixth Painlevé equation. Up to now, there are several generalizations of Frobenius manifold such as F -manifold by C. Hertling and Y.
Manin [10,9] and Saito structure (without metric) by C. Sabbah [31]. A. Arsie and P. Lorenzoni [1,23] showed that three-dimensional regular semisimple bi-flat F -manifolds are parameterized by solutions to (full-parameter) Painlevé VI equation, which may be 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 solutions to Painlevé V and IV equations.
The theory of linear differential equations on a complex domain is a classical branch of Mathematics. In recent years there has been great progress in this branch as a turning point with the introduction of the notions of middle convolution and rigidity index by N. M. Katz [18]. Especially T. Oshima developed a classification theory of Fuchsian differential equations in terms of their rigidity indices and spectral types [28,29,27]. ( [8] is a nice introductory text on the "Katz-Oshima theory".) Okubo systems play a central role in these developments (cf. [5,6,29,36]): a matrix system of linear differential equations with the form where T, B ∞ are constant square matrices, is an Okubo system if T is diagonalizable. If T is not necessarily diagonalizable, (1) is said to be a generalized Okubo system, which was studied by H. Kawakami [20,19] in order to generalize the middle convolution to linear differential equations with irregular singularities.
In this paper, we study isomonodromic deformations of a generalized Okubo system. We show that the space of variables of isomonodromic deformations of a generalized Okubo system can be equipped with a Saito structure (without a metric). (In the sequel, we abbreviate "Saito structure (without metric)" to "Saito structure" for brevity. In addition, "flat structure" in this paper stands for the same meaning as "Saito structure", c.f. [15, §1].) A major motivation of the present paper is Arsie-Lorenzoni's work [2,23], in which it was shown that three-dimensional regular bi-flat F -manifolds are parametrized by solutions to the Painlevé IV,V,VI equations. (It is proved in [3,22] that Arsie-Lorenzoni's bi-flat F -manifold is equivalent to Sabbah's Saito structure.) In [2], it has been left as an open problem how the remaining Painlevé equations (i.e. PIII, PII, PI) can be related with Saito structures. We give an answer to this problem in the present paper: we show that solutions to the Painlevé III and II are related with four-dimensional regular Saito structures. As for the Painlevé I, it can be related with isomonodromic deformations of a generalized Okubo system of rank seven (which is minimal rank), however it does not satisfy the regularity condition. For this reason, PI can not be treated in the framework of this paper. As another application, we study the initial value problem for regular Saito structures. L. David and C. Hertling [4] studied the initial condition theorem for regular F -manifolds and Frobenius manifolds. F -manifold is a lower structure of Saito structure.
The initial condition for a regular Saito structure consists of that for the underlying regular F -manifold and a data to specify a flat coordinate system and its weight (c.f. Corollary 3.13 and Remark 3.7). This paper is constructed as follows. In Section 2, we introduce the notion of extended generalized Okubo system as a completely integrable Pfaffian system extending a generalized Okubo system. We see that an extended generalized Okubo system is equivalent to an isomonodromic deformation of a generalized Okubo system (Lemma 2.3). We observe that an extended generalized Okubo system naturally induces a Saito bundle on a complex domain, which will be used in Section 3.2. In Section 3.1, we review the general theory of Saito structure. Particularly we introduce a potential vector field and the extended WDVV equation for a Saito structure. In Section 3.2, we show that the space of variables of an extended generalized Okubo system can be equipped with a Saito structure under some generic condition (Theorem 3.12). The arguments there closely follow [31, Chapter VII]: we find a condition for that the Saito bundle induced by an extended generalized Okubo system in Section 2 has a primitive section. In Section 4, we introduce Saito structures on the spaces of independent variables of solutions to the Painlevé equations. It is well known that each Painlevé equation (PI-PVI) can be derived from isomonodromic deformations of certain 2 × 2 system of linear differential equations. Kawakami [19,20] showed that any system of linear differential equations can be transformed into a generalized Okubo system. Using Kawakami's construction, we obtain a generalized Okubo system whose isomonodromic deformations are governed by solutions to each Painlevé equation. Then in virtue of Theorem 3.12, we see that generic solutions to the Painlevé VI,V,IV equations correspond to solutions to the three-dimensional extended WDVV equation satisfying some regularity condition (Theorem 4.6), which provides another proof of Arsie-Lorenzoni's result mentioned above. Moreover generic solutions to the Painlevé III and II equations correspond to solutions to the four-dimensional extended WDVV equation. In Section 5, we treat uniformly the Painlevé VI-II equations. In the previous section, we have seen that PIII and PII are related with four-dimensional regular Saito structures. The system of linear differential equations whose isomonodromic deformations are governed by PVI,PV,PIV can also be transformed into generalized Okubo systems of rank four. Then we see that generic solutions to PVI-PII correspond to solutions to the four-dimensional extended WDVV equation (Theorem 5.4). The well-known coalescence cascade of the Painlevé equations corresponds in our framework (except for PI) to the degeneration scheme of the Jordan normal forms of a square matrix of rank four: It is a rather unexpected aspect of the Painlevé equations that one can treat them (except PI) all together as just one system of differential equations. It seems to be an interesting problem to study analytic properties of Painlevé transcendents uniformly in terms of potential vector fields (see [14,15,16,17] for algebraic solutions to PVI). We will study this problem elsewhere. In Appendix A, we give a proof of Lemma 2.3. For this aim, we briefly review the theory of isomonodromic deformations of systems of linear differential equations with irregular singularities following Jimbo-Miwa-Ueno [12]. In Appendix B, following [19,20], we explain how to construct a generalized Okubo system from a given system of linear differential equations. This construction is used in Sections 4 and 5.
Acknowledgments The authors thank Professor C. Hertling for reading a preprint and giving useful comments on it and Professor M. Noumi for useful discussions. This work was partially supported by JSPS KAKENHI Grant Numbers 25800082, 17K05335.

Generalized Okubo system
For N ∈ N, let T and B ∞ be N × N-matrices. The system of ordinary linear differential equations where I N denotes the identity matrix of rank N, is said to be an Okubo system if the matrix T is diagonalizable (cf. [26]), and said to be a generalized Okubo system if T is not necessarily diagonalizable (cf. [20]). Let U be a simply-connected domain in C N . We extend (3) to a completely integrable Pfaffian system of the form on P 1 × U, or equivalently an integrable meromorphic connection on the trivial bundle of rank N over P 1 × U: where z is a non-homogeneous coordinate of P 1 andΩ is an N × N matrix valued 1-form on U satisfying the conditions below. We put the following assumptions on (4): (A1) T (resp.Ω) is an N × N matrix whose entries are holomorphic functions (resp. holomorphic 1-forms) on U.
(A3) The matrix T is generically regular. Namely, let T have the Jordan normal form where J k is a Jordan block of rank m k : and m k ∈ N, 1 ≤ k ≤ n are subject to m 1 + · · · + m n = N, and put Then δ H red is not identically zero on U.
By the assumption (A3), there exists an invertible matrix P whose entries are holomorphic functions on an open set W ⊂ U \ {δ H red = 0} such that where Then we assume also the following (A4): (A4) n k=1 dz k,0 ∧ · · · ∧ dz k,m k −1 = 0 at generic points on W . This means that (z k,0 , . . . , z k,m k −1 ) 1≤k≤n can be taken as a coordinate system on a sufficiently small open set in W .
Note that Z k is written as Lemma 2.1. Assume that B ∞ is invertible. Then the Pfaffian system (4) is completely integrable if and only if the matrices T,Ω, B ∞ satisfy the following system of equations dT +Ω + [Ω, In particular, T andΩ are simultaneously block diagonalizable: for an invertible matrix P , it holds that P −1 T P = Z 1 ⊕ · · · ⊕ Z n , P −1Ω P = −(dZ 1 ⊕ · · · ⊕ dZ n ).
Proof. The assertions in the lemma were proved in [15] in the case where T in (4) is diagonalizable. We prove the lemma in the general case by constructing a confluence process from the case where T is diagonalizable.
Then we readily see from which it follows that where P is the same as in (7). Since Z ε is diagonal, it is known in [15] that the Pfaffian system is completely integrable if and only if T ε ,Ω ε , B ∞ satisfy (10),(11) andΩ ε is represented as From the limit of (16) and (13) as ε → 0, we have and we see that T,Ω, B ∞ satisfy (10), (11).
(ii) If a generalized Okubo system (3) is given on P 1 × {p 0 } for p 0 ∈ U \ {δ H red = 0}, then there exists a unique extended generalized Okubo system on P 1 × W which coincides with the given (3) restricted on Proof. The assertion (i) is proved in Appendix A. The assertion (ii) follows from the existence and uniqueness of solutions for initial conditions of the isomonodromic deformation shown in [12].
An extended generalized Okubo system naturally induces a Saito bundle as follows. Let us recall the definition of Saito bundle following [4,31]: , such that the following conditions are satisfied: Returning to our setting, let C N (U) be a trivial bundle of rank N on a domain U and (e 1 , . . . , e N ) t be a basis of sections of C N (U). Let ∇ (0) be the trivial connection on C N (U) defined by ∇ (0) (e i ) = 0 (i = 1, . . . , N). Then, for an extended generalized Okubo system (4), Lemma 2.1 implies that the 5-tuple (C N (U), ∇ (0) ,Ω, T, −B ∞ ) defines a Saito bundle on U.

Review on Saito structure (without metric)
In this section, we review Saito structure (without metric) introduced by C. Sabbah [31]. Proofs of many of statements in this section are found in the literature ( [31,15,21]). In the sequel, we abbreviate Saito structure (without metric) to Saito structure for brevity. (a) A meromorphic connection ∇ on the bundle π * T M on P 1 × M defined by is integrable, where π is the projection π : P 1 ×M → M and z is a non-homogeneous coordinate of P 1 , The Higgs field Φ is said to be symmetric if the product ⋆ is commutative and associative. The condition Φ e = Id in Definition 3.1 (b) implies that the field e is the unit of the product ⋆. So we can introduce on Θ M the structure of associative and commutative O M -algebra with a unit.
Since the connection △ is flat and torsion free, we can take a flat coordinate system (t 1 , . . . , t N ) such that △ (∂ t i ) = 0, 1 ≤ i ≤ N, (at least) on a simply-connected open set U of M. We take a flat coordinate system (t 1 , . . . , t N ) on U and assume the following conditions: In this paper, a function f ∈ O M (U) is said to be weighted homogeneous with a weight w(f ) ∈ C if f is an eigenfunction of the Euler operator: and introduce the following matrices: In the following, we assume that −Φ(E) (or equivalently T ) is generically regular on U (cf. (A3) in Section 2).
Remark 3.2. By the assumption that −Φ(E) is generically regular, we can take a coordinate system (−z k,0 , . . . , −z k,m k −1 ) 1≤k≤n as (A4) in Section 2, which is called a canonical coordinate system. The canonical coordinate system for regular F -manifolds are constructed by L. David and C. Hertling [4]. The product ⋆ is written in a simple form with respect to the canonical coordinate system. Indeed the product is represented with is integrable if and only if T ,Φ and B ∞ are subject to the following system of equations Lemmas 3.2 and 3.3 imply that a Saito structure induces an extended generalized Okubo system and thus a Saito bundle (C N (U), △ ,Φ, T , −B ∞ ) by taking a flat coordinate system on U.
Remark 3.3. The meromorphic connection (19) is written as the following Pfaffian system with respect to the flat coordinate system: The system of equations (22), (23) is equivalent to the integrability condition of (24). The system of ordinary linear differential equations has an irregular singular point of Poincaré rank 1 at z = 0 and a regular singular point at z = ∞, which is called a Birkhoff normal form. A Birkhoff normal form can be transformed into a generalized Okubo system using a Fourier-Laplace transformation.
Lemma 3.4. There is a unique matrix C whose entries are in O M (U) such that and that each matrix entry C ij of C is weighted homogeneous with w(C ij ) = 1 − w i + w j .
Lemma 3.5. Let (t 1 , . . . , t N ) be a flat coordinate system of a Saito structure. Then it [21]). There is a unique N-tuple of holomorphic func- ∂t i , and that g j is weighted homogeneous with w(g j ) = 1 + w j . The vector g (or precisely the vector field is called a potential vector field. Remark 3.4. It is readily found that the potential vector field g = (g 1 , . . . , g N ) is explicitly given by Proposition 3.7. The potential vector field g = (g 1 , . . . , g N ) is a solution to the following system of nonlinear differential equations: Definition 3.8. The system of non-linear differential equations (26)-(28) for a vector g = (g 1 , . . . , g N ) is called the extended WDVV equation.
Conversely, starting with a solution to (26)- (28), it is possible to reconstruct a Saito structure. Proposition 3.9. Take constants w j ∈ C, 1 ≤ j ≤ N satisfying w i − w j ∈ Z and w N = 1 and assume that g = (g 1 , . . . , g N ) is a holomorphic solution to (26)-(28) on a simplyconnected domain U in C N . Then there is a Saito structure on U which has (t 1 , . . . , t N ) as a flat coordinate system.
satisfies the conditions (a), (b) in Definition 3.1 and g becomes its potential vector field.

Saito structure on the space of isomonodromic deformations of a generalized Okubo system
We consider an extended generalized Okubo system (4) with the same assumptions as in Section 2. We show that the space U of variables of (4) is equipped with a Saito structure under some generic condition. The arguments below closely follow [31,Chapter VII]. In a general setting, for a Saito bundle (V, on a complex manifold M, a ∇ V -horizontal section ω of V is said to be a primitive section if it satisfies the following conditions: Thanks to [31, Chapter VII, Theorem 3.6], if there is a primitive section ω, we can introduce a Saito structure ( △ , Φ, e, E) on M via ϕ ω : Returning to our setting, we consider an extended generalized Okubo system (4) defined on P 1 × U, where U is a domain on C N . For a while, we treat (4) on an appropriate smaller domain W ⊂ U \ {δ H red = 0} so that we can take an invertible matrix P such that P −1 T P = Z 1 ⊕ · · · ⊕ Z n , P −1Ω P = −dZ 1 ⊕ · · · ⊕ dZ n .
So far we have treated (4) on W outside of {δ H red = 0}. From now on, we consider (4) on U including {δ H red = 0}.
When the condition (30) is satisfied, the set of variables t j := −(λ j −λ N +1) −1 T N j = C N j , 1 ≤ j ≤ N gives a flat coordinate system.
Proof. In virtue of Lemmas 3.10 and 3.11, the theorem holds on W ⊂ U \ {δ H red = 0}.
We also see that {t j = C N j } is a flat coordinate system by Lemma 3.5. Then (  Applying Theorem 3.12, we can solve the initial value problem on regular Saito structures for generic initial conditions. Corollary 3.13. The following two assertions hold: (i) Let S be a constant N ×N matrix that is regular in the sense of (A3) in Section 2, and R be a constant N ×N matrix whose eigenvalues {λ 1 , . . . , λ N } satisfy λ i −λ j ∈ Z\{0} for i = j and v be one of eigenvectors of R. For (S, R, v) and a point p 0 ∈ C N , if S is generic (the meaning of that "S is generic" is clarified in the proof ), there exists a regular Saito structure on a neighborhood of p 0 uniquely up to isomorphisms.
(ii) Let ( △ , Φ, E, e) be a regular Saito structure on a domain U and p 0 ∈ U \ {δ H red = 0} for H(z) = (z · id T U − T ) (see (A3) in Section 2 for the definition of δ H red ). Then the Saito structure ( △ , Φ, E, e) is uniquely determined up to isomorphisms by Proof. For a given (S, R, v) satisfying the assumption in (i), let λ N be the eigenvalue of v. Take an invertible matrix G such that its N-th column coincides with v and such that G −1 RG = diag(λ 1 , . . . , λ N ). We consider the following generalized Okubo system: where we put T := G −1 SG and B ∞ := diag(λ 1 , . . . , λ N ). Then in virtue of Lemma 2.3, there exists a unique extended generalized Okubo system on P 1 × W determined by (31), where W ⊂ U \ {δ H red = 0} is a neighborhood of p 0 . If the extended generalized Okubo system satisfies the condition (30) of Theorem 3.12 (which is the meaning of "generic condition for S" mentioned in the assertion (i)), there exists a unique Saito structure on W determined by (S, R, v). We prove (ii). The unit field e| p 0 ∈ T p 0 X at p 0 is an eigenvector of △ (E)| p 0 belonging to the eigenvalue w N = 1 (i.e. △ e (E) = e), and thus e| p 0 is recovered from △ (E)| p 0 . Then, from (i), we see that the Saito structure ( △ , Φ, E, e) is uniquely determined by Remark 3.7. Saito structure is an upper structure of F -manifold introduced by C. Hertling and Y. Manin [10,9]. The initial condition theorem for regular F -manifolds was studied by L. David and C. Hertling [4]. In particular, they proved that an F -manifold is uniquely determined up to isomorphisms by −Φ(E)| p 0 .

Potential vector fields associated with solutions to the Painlevé equations
In this section, applying the results in the previous section, we introduce potential vector fields associated with solutions to the (classical) Painlevé equations.
In [15], the following was proved: 15]). In the case of N = 3, there is a correspondence between generic solutions to the sixth Painlevé equation PVI and generic solutions to the extended WDVV equation satisfying the additional condition Remark 4.1. A. Arsie and P. Lorenzoni [1,23] showed that three-dimensional regular semisimple bi-flat F -manifolds is parameterized by solutions to the Painlevé VI equation.
Regarding the Painlevé V equation, we obtain the following result as a consequence of the arguments in the present paper: Proof. According to [13,19], the Painlevé V equation is derived from the isomonodromic deformation of the system of linear differential equations which has regular singular points at z = 0, ∞ and an irregular singular point of Poincaré rank 1 at z = 1: where A 0 , A 1 , A 2 are 2 × 2 matrices independent of z. On (37), we assume the following conditions: (i) the exponent matrix at z = 0 is given by T (iii) a fundamental system of formal solutions at z = 1 is given as where Under these assumptions, we find that the matrices A 0 , A 1 , A 2 are written as where t is a deformation parameter, p, q are unknown functions for the isomonodromic deformation, θ 0 , θ 1 , θ ∞ 1 , θ ∞ 2 are constant parameters satisfying θ 0 + θ 1 + θ ∞ 1 + θ ∞ 2 = 0 and u is an overall parameter. The Painlevé V equation is equivalent to the Hamiltonian system with the Hamiltonian Indeed, after the canonical transformation λ satisfies the Painlevé V equation Now we prove the proposition. At the first step, we start from a solution to the extended WDVV equation (32)-(34) satisfying the condition (36). Then we have a 3 × 3 extended generalized Okubo system (4) with The condition (36) implies that there is an invertible matrix P such that Then, by the procedure explained in Appendix B, we can reduce the extended generalized Okubo system to an isomonodromic deformation of the following 2 × 2 system of linear differential equations: Change the variablez → z = (z −z 2,0 )/(z 1,0 −z 2,0 ). Then (44) is changed to (37) satisfying the conditions (i)-(iii). Hence we obtain a solution to (40), which is equivalent to a solution to PV. Particularly we have the following correspondence between parameters: for the matrix P in (43), put B := −P −1 B ∞ P . Then Next we start from an isomonodromic deformation of the system (37) with the conditions (i)-(iii) (which is equivalent to giving a generic solution to the Painlevé V equation). The system (37) can be transformed into a generalized Okubo system of rank three applying the procedure explained in Appendix B (cf. [19]): where Note that S V is regular in the sense of (A3) in Section 2. Change the variables z →z = (z 1,0 − z 2,0 )z + z 2,0 and t → z 1,1 = (z 1,0 − z 2,0 )t. Diagonalizing C V by applying a gauge transformation to (45), we obtain from S V a matrix T satisfying (43). If the resulting matrix T satisfies the condition (30) in Theorem 3.12 (which is a generic condition), we obtain a flat coordinate system and a potential vector field satisfying the condition (36).
We can obtain similar results on the remaining Painlevé equations except for Painlevé I: Proof. The Painlevé IV equation is derived from the isomonodromic deformation of the following 2 × 2 system of linear differential equations which has a regular singular point at z = ∞ and an irregular singular point of Poincaré rank 2 at z = 0: The Painlevé IV equation is equivalent to the Hamiltonian system with the Hamiltonian We start from a solution to the extended WDVV equation (32)-(34) satisfying the condition (46). In a way similar to the PV case, we can reduce the 3 × 3 generalized Okubo system to the 2 × 2 system (47), and thus we have a solution to the Painlevé IV equation.
Conversely, starting from an isomonodromic deformation of the 2 × 2 system (47), we can construct a 3 × 3 generalized Okubo system similarly to the case of PV (cf. [19,20]): where Noting that S IV satisfies the regularity condition which is equivalent to (46), we obtain a potential vector field satisfying the condition (46) by a manner similar to the PV case.
with the constraint on the weight w 1 = w 2 , w 3 = w 4 and the additional condition Proof. The Painlevé III equation is derived from the isomonodromic deformation of the following 2 × 2 system of linear differential equations which has irregular singular points of Poincaré rank 1 at z = 0, ∞: where The Painlevé III equation is equivalent to the Hamilton system with the Hamiltonian Note that (54) has no regular singular point, whereas any generalized Okubo system necessarily has a regular singular point at ∞. We change the variables z → ξ = z z−1 and Y → Z = (ξ − 1) θ ∞ 2 Y in order to add a regular singularity at ∞ to (54). Then (54) is which has irregular singular points of Poincaré rank 1 at ξ = 0, 1 and a regular singular point at ξ = ∞. The residue matrix of (55) at ξ = ∞ reads θ ∞ 2 I 2 . We start from a solution to the extended WDVV equation (50)-(52) satisfying the condition (53) with the constraint w 1 = w 2 , w 3 = w 4 . Then we have a 4 × 4 extended generalized Okubo system with B ∞ = diag(w 1 , w 1 , w 3 , w 3 ). Applying the procedure explained in Appendix B, we can reduce the 4 × 4 generalized Okubo system to the 2 × 2 system (55), which is equivalent to (54). Hence we obtain a solution to the Painlevé III equation.
Conversely, we start from an isomonodromic deformation of the 2 × 2 system (54), which is equivalent to an isomonodromic deformation of (55). We can construct from the 2 × 2 system (55) a 4 × 4 generalized Okubo system by the procedure explained in Appendix B, which is determined by the following data consisting of three matrices {S III , G III , B ∞ }: Note that S III satisfies the regularity condition which is equivalent to the condition (53), and B ∞ satisfies the constraint w 1 = w 2 , w 3 = w 4 . Hence we obtain a potential vector field satisfying the desired conditions.
Proof. The Painlevé II equation is derived from isomonodromic deformations of the following 2 × 2 system of linear differential equations which has an irregular singular point of Poincaré rank 3 at z = ∞: where

The Painlevé II equation is equivalent to the Hamiltonian system with the Hamiltonian
By a reason similar to PIII, we change the variables z → ξ = 1/z and Y → Z = ξ θ ∞ 2 Y . Then (57) is changed into which has an irregular singular point of Poincaré rank 3 at ξ = 0 and a regular singular point at ξ = ∞. The residue matrix of (59) at ξ = ∞ reads −θ ∞ 2 I 2 . It is similar to PIII that, starting a solution to the extended WDVV equation (50)-(52) satisfying the condition (56) with the constraint w 1 = w 2 , w 3 = w 4 , we obtain a solution to the Painlevé II equation.
Conversely, we start from an isomonodromic deformation of the 2 × 2 system (57), which is equivalent to an isomonodromic deformation of (59). We can construct from the 2 × 2 system (59) a 4 × 4 generalized Okubo system by the procedure explained in Appendix B, which is determined by the following data consisting of three matrices {S II , G II , B ∞ }: Note that S II satisfies the regularity condition which is equivalent to the condition (56), and B ∞ satisfies the constraint w 1 = w 2 , w 3 = w 4 . Hence we obtain a potential vector field satisfying the desired conditions.
We may summarize the three-dimensional case as follows.
Theorem 4.6. There is a correspondence between generic solutions satisfying the regularity condition to the extended WDVV equation and generic solutions to the Painlevé equations PVI,PV,PIV, where "the regularity condition" means that g satisfies one of (35), (36), (46). The first Painlevé equation PI is derived from the isomonodromic deformation of the following 2 × 2 system of linear differential equations: where which has an irregular singular point of Poincaré rank 5/2 at z = ∞. PI is equivalent to the Hamiltonian system with the Hamiltonian H I = p 2 − q 3 − tq.
Similarly to the case of PII, we change the variables of (63) as z → ξ = 1/z and We see that (64) is transformed into the following 7 × 7 generalized Okubo system by the procedure explained in Appendix B (which is a realization of (64) as a generalized Okubo system of minimal size): We notice that S I is not regular and thus we can not treat PI in the framework of the present paper.

Unified treatment of the Painlevé equations
In the previous section, we related each of the Painlevé equations with an extended generalized Okubo system, which is of minimal rank. As for PVI, PV and PIV, it is possible to relate them to extended generalized Okubo systems of rank four.
Proof. PVI is derived from the isomonodromic deformation of the following 2 × 2 system of linear differential equations: where we assume det A 1 = det A 2 = det A 3 = 0 without loss of generality. We add a regular singularity to (66): for t 2 ∈ C\{0, 1}, we change the variables x → ξ = t 2 x x+t 2 −1 and where we put t 1 := t t 2 t+t 2 −1 . It is apparent that (67) has regular singular points at ξ = 0, 1, t 1 , t 2 , ∞, and the residue matrix at ξ = ∞ reads λI 2 .
We start from a solution to the extended WDVV equation (50)-(52) satisfying the condition (65) with the constraint w 1 = w 2 , w 3 = w 4 . Then we have a 4 × 4 extended Okubo system with B ∞ = diag(w 1 , w 1 , w 3 , w 3 ). Applying the procedure explained in Appendix B, we can reduce the 4 × 4 generalized Okubo system to the 2 × 2 system (67), with the change of variables Hence we obtain a solution to the Painlevé VI equation. Conversely, we start from an isomonodromic deformation of the 2 × 2 system (67), which is equivalent to an isomonodromic deformation of (66). We can construct from the 2 × 2 system (67) a 4 × 4 generalized Okubo system by the procedure explained in Appendix B, particularly we find Note that S VI ′ satisfies the regularity condition which is equivalent to the condition (65), and B ∞ satisfies the constraint w 1 = w 2 , w 3 = w 4 . We obtain a potential vector field satisfying the desired conditions.
Remark 5.1. The correspondence between particular 4-dimensional Frobenius manifolds and generic solutions to a one-parameter family of the Painlevé VI equation was treated by S. Romano [30] (in a somewhat different context).
We obtain similar results on PV and PIV: Proposition 5.2 (PV). In the case of N = 4, there is a correspondence between generic solutions to the Painlevé V equation and generic solutions to the extended WDVV equation (50)-(52) with the constraint w 1 = w 2 , w 3 = w 4 and the additional condition Consequently we obtain the following theorem.
Here we remark that Ω, Θ (µ) are obtained from A, G (µ) by a rational procedure which is described by [12, (3.14) and (3.16)]. Now we step forward to proving Lemma 2.3. Consider an extended generalized Okubo system with the same assumptions as in Section 2: The system of differential equations in the z-direction of (81) is rewritten to the form of (74): taking an invertible matrix P such that P −1 T P = Z 1 ⊕ · · · ⊕ Z n , (82) is written as where m k −1 j=0 A k,−j (z − z k,0 ) −j−1 = −P O ⊕ · · · ⊕ (zI m k − Z k ) −1 ⊕ · · · ⊕ O P −1 B ∞ .
We remark that z = ∞ is a regular singular point with the residue matrix B ∞ . Lemma 2.3 follows from the following lemma: Lemma A.1. The extended generalized Okubo system (81) is equivalent to the completely integrable Pfaffian system consisting of (83) and (79).
Proof. The matrix valued 1-form Ω(z) in (79) is uniquely characterized by the following conditions: (i) Each entry of n k=1 (z − z k,0 ) m k Ω(z) is a polynomial in z. B Construction of a generalized Okubo system from a system of linear differential equations In this appendix, we explain following [19,20] how to construct a generalized Okubo system from a given linear differential equation of the type of (74) with a regular singular point at ∞. This construction is used in Sections 4 and 5.
As the first step, we start from a generalized Okubo system with assumptions (A2) in Section 2 and observe that the rank of (84) can be reduced by the following procedure. Let B ∞ = diag(λ 1 , . . . , λ N ) and suppose that λ m+1 = · · · = λ N , HenceỸ = t (y 1 , . . . , y m ) satisfies the following m × m matrix differential equation: Next, we give a construction in the opposite direction. We start from an arbitrary m × m matrix differential equation: where we assume that (87) has a regular singular point at z = ∞ and thatR := − n k=1 A (0) k is a diagonal matrix:R = diag(λ 1 , . . . , λ m ). Our goal is to transform (87) into a generalized Okubo system. Find a natural number N, an m × N-matrix B, an N × m-matrix C and an N × N-matrix S in the Jordan normal form such that If we obtain such matrices, then is a generalized Okubo system, where G = CR −1C , B ∞ = diag(λ 1 , . . . , λ m , 0, . . . , 0).
In particular, the data consisting of the matrices {S, G, B ∞ } determines the generalized Okubo system (88). We note that the Jordan normal form S for (87) is unique provided that the size of S is minimal. Then B and C is unique up to (B, C) ∼ (Bh −1 , hC) where h ∈ Stab(S) ( [35]). This implies the uniqueness of G up to left multiplication by Stab(S) and right multiplication by Stab(B ∞ ).