Four-Dimensional Painlev\'e-Type Equations Associated with Ramified Linear Equations III: Garnier Systems and Fuji-Suzuki Systems

This is the last part of a series of three papers entitled"Four-dimensional Painlev\'e-type equations associated with ramified linear equations". In this series of papers we aim to construct the complete degeneration scheme of four-dimensional Painlev\'e-type equations. In the present paper, we consider the degeneration of the Garnier system in two variables and the Fuji-Suzuki system.


Introduction
This is the last part of a series of three papers on four-dimensional Painlevé-type equations associated with ramified linear equations. By the term "Painlevé-type equations" we mean Hamiltonian systems which describe isomonodromic deformations of linear equations. The isomonodromic deformation is the deformation of a linear differential equation which do not change its "monodromy data" (see, for example, [9]), and it is known that isomonodromic deformation equations can be written in Hamiltonian form. In this terminology, the classical Painlevé equations are Painlevé-type equations with two-dimensional phase space.
The classical Painlevé equations are non-linear ordinary differential equations which were discovered by Painlevé [26] and Gambier [4]. Originally they were classified into six equations and are often denoted by P I , P II , . . . , P VI . However, from a geometric viewpoint, it is natural to classify them into eight equations [27]. More precisely, the third Painlevé equation P III is divided into three cases P III(D 6 ) , P III(D 7 ) , and P III(D 8 ) . The so-called third Painlevé equation is then P III(D 6 ) .
The standard linear equations associated with the classical Painlevé equations are given by certain second order single linear equations, or equivalently, by first order 2 × 2 systems. Here we review the classification of the classical Painlevé equations in terms of associated linear equations.
It is well-known that the classical Painlevé equations admit degeneration. We use the term degeneration in the following sense. Suppose a differential equation E has some parameter ε. When the equation E tends to another equation E as ε tends to 0, we say that E degenerates to E . The following scheme is the well-known degeneration scheme among the six Painlevé equations: The number in each box is the "singularity pattern" of an associated linear equation, which has information on the Paincaré ranks of the singular points of the linear equation. In this case, the linear equations are well characterized by their singularity pattern.
Here we point out that • this scheme lacks the third Painlevé equations of type D (1) 7 and D (1) 8 , • from the viewpoint of associated linear equations, the degeneration H II → H I is distinguished from the others. Namely, the other degenerations correspond to the "confluence of singular points", while the degeneration H II → H I corresponds to the "degeneration of an HTL canonical form". (11)(1), (11)(1) Symbols such as H 2+1+1+1 Gar stand for the Hamiltonians for four-dimensional Painlevé-type equations. The explicit expressions for them are given in Section 3.

Notions on linear dif ferential equations
In this section we recall some notions on linear differential equations.

HTL canonical forms
Consider a system of linear differential equations A change of the dependent variable Y = P Z with an invertible matrix P = P (x) yields the following system:

The transformation
A(x) → A P (x) := P −1 A(x)P − P −1 P is called the gauge transformation by P .
Remark 2.3. In this series of papers, the matrix Θ is always diagonalizable.
The table of the HTL forms (represented by the above formula) at all singular points is called the Riemann scheme of a linear equation.
Concerning the computation of HTL forms, the following two theorems are fundamental.
Theorem 2.4 ( [33]). For any there exists a formal power series P = P (z) such that When no two different eigenvalues of A 0 differ by an integer, we can choose the gauge so that Theorem 2.5 (block diagonalization [33]). Let A(z) be where the eigenvalues of A 0 are assumed to be λ 1 , . . . , λ n . Without loss of generality, we can assume that A 0 is in Jordan canonical form where J k (λ k ) is a direct sum of Jordan blocks with the eigenvalue λ k . Then there exists a formal power series P = P (z) such that where B k 0 = J k (λ k ).

Spectral types
We have introduced the notion of singularity pattern to represent the singularity of a linear system. However, if the rank of a linear system is greater than two, the singularity pattern is too rough to describe the singularity of the linear system sufficiently since it only has information of the Poincaré ranks. In order to describe the singularity of a linear system in more detail, we need the notion of spectral type of linear equations.
Spectral types are defined through HTL forms.
• In the unramified case, the spectral type is defined to be the (tuple of) "refining sequence of partitions (RSP)".
• In the ramified case, the spectral type consists of "copies" of RSPs.
Thus we begin with the unramified case.
Four-Dimensional Painlevé-Type Equations Associated with Ramified Linear Equations III 9 2.3.1 Spectral types of unramif ied irregular singularities Consider a system of linear differential equations whose coefficient matrix is Now suppose that A 0 is diagonalizable. Applying the block diagonalization (see Theorem 2.5) to (2.3), we see that the leading terms B k 0 (k = 1, . . . , n) are all scalar matrices. We focus on the next B k 1 in each block. Suppose that all the B k 1 (k = 1, . . . , n) are also diagonalizable. Then diagonalize them by constant gauge transformations and apply Theorem 2.5 to each block. Thus each block decomposes into smaller direct summands again. We note that the first two terms of each direct summands thus obtained are scalar matrices.
In general, if B r−j+1 is diagonalizable in a direct summand We denote an RSP in the following way. Second, put the numbers that are grouped together in the central partition in parentheses: Finally, put the numbers that are grouped together in the leftmost partition in parentheses: ((11)(1))((11))( (1)).

Spectral types of ramif ied irregular singularities
In general, non-semisimple matrices may appear in a sequence of block diagonalizations. Such a case can be reduced to the above semisimple case by "shearing transformations". Here we point out that, usually, the non-semisimplicity implies the ramifiedness of a singular point. A shearing transformation is a gauge transformation by a diagonal matrix, which is typically of the form where s is a positive rational number. The aim of the shearing transformation is to make nonsemisimple coefficient matrices semisimple by repeating gauge transformations of the above kind [8,32]. Instead of describing shearing transformations and constructions of HTL forms in the ramified case in full generality, we demonstrate a construction of an HTL form using the linear system (3.5). A general method of constructing HTL forms can be found in [33] (see also [14]).
First we change the dependent and independent variables as z = 1/x, Y = diag(1, −1, 1)Z, we rewrite (3.5) as follows We consider the singular point z = 0.
Four-Dimensional Painlevé-Type Equations Associated with Ramified Linear Equations III 11 Now we perform shearing transformations. Let S 1 be the diagonal matrix diag(1, z 1/3 , z 2/3 ). Then we have How to find the rational number s of the shearing matrix is described in [33]. In this case, the dimension of the centralizer of the leading coefficient matrix of A S 1 (z) is less than that of A(z).
In fact, let G 1 be the following matrix Then we have the following Jordan canonical form of the leading matrix of A S 1 (z): Next, let S 2 = diag(1, z 1/3 , z 2/3 ). Then we have Note that the leading matrix of A S 1 G 1 S 2 (z) is diagonalizable. Indeed, the following matrix Since the leading matrix has three distinct eigenvalues, we can remove the off-diagonal entries by virtue of Theorem 2.5. That is, there exits a matrix P = P (z) whose entries are formal Laurent series in z 1/3 such that A S 1 G 1 S 2 G 2 P is diagonal: Then, by virtue of Theorem 2.4, we can truncate A S 1 G 1 S 2 G 2 P (z) after the principal part by a certain diagonal gauge transformation. Furthermore, by the scalar gauge transformation by z −2/3 , we can cancel the term −2/3 z . In this way, we can obtain the HTL form of (3.5) Together with the HTL form at x = 0 (this can be easily seen), we obtain the Riemann scheme of the system (3.5): We briefly describe the feature of an HTL form (at a ramified irregular singularity) here. See [1] for a precise statement. Suppose the HTL form of A(z) ∈ M m (C((z))) has as its direct summand. Then the HTL form of A(z) also has which is the orbit of (2.5) under the action z of a cyclic group, as its direct summands.
Let S be the RSP corresponding to (2.5). We denote the collection of S and its d − 1 copies by S d . Then an HTL form is generally represented as S 1 d 1 . . . S k d k where S 1 , . . . , S k are RSPs; we call this the spectral type of an HTL form. The spectral type at a singular point of a linear system is defined as the spectral type of the HTL form at the point. The tuple of the spectral types at all singular points is called the spectral type of the equation.

Degeneration of HTL canonical forms
Suppose a system of linear equations has some parameter, say ε. When we take the limit ε → 0, usually with a gauge transformation by some matrix which depends on ε and is independent of x, an HTL form of the linear system at some singular point may change. We call this situation a degeneration of an HTL form.
In the two dimensional case, the standard linear systems associated with classical Painlevé equations are 2 × 2, and degenerations of HTL forms are realized by the degenerations of the Jordan canonical forms of the coefficient matrices of the leading terms at irregular singular points. However, when the rank of a linear system is greater than two, a degeneration of an HTL form does not necessarily correspond to a degeneration of a Jordan canonical form. We can see such degenerations in the degeneration schemes of the Sasano system and the Fuji-Suzuki system. Here we take a sequence of degenerations (11) 1) as an example.
Before going into the details, we look at the following simple example.
Example 2.9. Let A(x) be a 2 × 2 matrix of the form Applying the gauge transformation by S = diag(1, x 1/2 ) to (2.6), we have When a (1) 21 = 0, the leading coefficient of A S (x) is a diagonalizable matrix with eigenvalues ± a (1) 21 . This means that the singular point x = 0 of the system of linear differential equations corresponding to (2.6) is an irregular singular point of Poincaré rank 1/2. We can see that the HTL form at x = 0 is by diagonalizing the leading matrix. When a (1) 21 = 0, usingS = diag(1, x) instead of the above S, we have the different HTL form (we omit the details). This example implies that if the leading matrix is a Jordan canonical form whose (1, 2)-entry is non-zero, then whether the (2, 1)-entry of the subsequent matrix is zero or not is meaningful. Now we consider the degeneration (11)(1), (11)(1) → (1) 2 1, (11)(1). The linear system of the spectral type (11)(1), (11)(1) is given by (A.1). Note that the leading matrix A 0 at the irregular singular point x = ∞ is diagonalizable. The degeneration of the HTL form at x = ∞ is caused by the degeneration of the Jordan canonical form of A 0 : In fact, changing the variables of (A.1) as in Appendix A.2, we have the following new coefficient matrices HereÃ 0 is diagonalizable provided that ε is not equal to 0, and it degenerates to a nilpotent matrix when ε tends to 0. We note that the (2, 1)-entry of lim ε→0Ã1 is not equal to 0, and thus we have the linear system (3.4) of the spectral type (1) 2 1, (11)(1) by ε → 0.
Remark 2.10. It is easy to obtain the HTL form at x = ∞, which corresponds to (1) 2 1, of (3.4). In the same manner as Example 2.9, the shearing at x = ∞ can be done by the matrix On the other hand, the degeneration of a HTL form (1) 2 1 → (1) 3 does not correspond to the degeneration of a Jordan canonical form. Let us see the degeneration (1) 2 1, (11)(1) → (1) 3 , (11)(1). In the course of the degeneration, the Jordan canonical form of the leading matrix stays unchanged. Instead, the (2, 1)-entry of the subsequent matrix goes to zero. In fact, changing the variables of (3.4) as in Appendix A.2, we have where G = diag(t, 1, 1) (we have omitted the expression of G −1 A 2 G). The limit ε → 0 causes the degeneration of the HTL form (1) 2 1 → (1) 3 . In this way, we obtain the system (3.5). The construction of the HTL form at x = ∞ of (3.5) has been given in Section 2.3.2.
We determine the possibility of degeneration as follows. For example, (1) 2 1 is a direct sum of two direct summands (see the Riemann scheme of (1) 2 1, (11)(1)) Four-Dimensional Painlevé-Type Equations Associated with Ramified Linear Equations III 15 and θ ∞ 2 /z. From this, we expect that it is possible to take a limit t → 0 and indeed this corresponds to the degeneration (1) 2 1 → (1) 3 . On the other hand, (1) 3 itself consists of a single Galois orbit, see (2.4). Thus we conclude that (1) 3 does not admit degeneration.
Let us make a remark on degeneration of HTL forms. We do not consider the degeneration of the HTL form (2)(1) since the degeneration of (2)(1) does not preserve the number of accessory parameters.
Remark 2.11. Accessory parameters of a linear system are free parameters remaining in the linear system when the Riemann scheme is fixed. The number of accessory parameters of a linear system coincides with the dimension of the phase space of the corresponding Painlevétype equation.
To see this, for example, suppose that the HTL form (2)(1) of the (2)(1), 111, 111-system degenerates. Then the degenerated linear system has the following form: By a direct calculation, we find that the number of accessory parameters of the above system is six. In fact, there is a Fuchsian equation with spectral type 21, 111, 111, 111, which has six accessory parameters. The system (2.7) turns out to be the degenerated system of the 21, 111, 111, 111-system. The same argument applies to the degeneration of ((11))((1)) of the ( (11))((1)), 111-system.

Lax pairs of degenerate FS and Garnier systems
The Garnier systems and the Fuji-Suzuki systems are non-linear differential equations, which are regarded as generalizations of the Painlevé equations. The Garnier system in N variables was derived as the isomonodromic deformation equation of a second order Fuchsian equation with N + 3 singular points [5]. The Fuji-Suzuki systems were originally derived from the Drinfeld-Sokolov hierarchy by similarity reductions [3].
In Section 3.1, we present the Riemann schemes, Lax pairs, and corresponding Hamiltonians of degenerate Garnier systems. In Section 3.2, we present similar data of degenerate Fuji-Suzuki systems.

degenerate Garnier systems
The Garnier systems were originally derived from a second order single Fuchsian equation with N + 3 singular points. However, unlike the original study by Garnier, we adopt first order systems concerning linear equations. In [28], the Garnier system in N variables was derived from the first order 2 × 2 system of the form When N equals 2, the Painlevé-type equation corresponding to (3.1) has a four-dimensional phase space. In [15], confluences from this linear system were considered. The degeneration of the Garnier system in two variables was considered by Kimura [17]. He treated mainly the confluence of singular points of associated linear equations, and he obtained the degenerated Garnier systems with the singularity pattern 2 + 1 + 1 + 1, 3 + 1 + 1, 2 + 2 + 1, 3 + 2, 4 + 1, 5, and 9/2. Kawamuko [16] further considered the degeneration of HTL canonical forms and obtained eight degenerate Garnier systems.
In this subsection, we give the Riemann schemes, Lax pairs, and Hamiltonians for degenerate Garnier systems associated with ramified linear equations. Although all the Hamiltonians in this subsection are equivalent to those in [17,16], we recalculated them. The following is the list of Hamiltonians for the degenerate Garnier systems associated with ramified linear equations: Gar,t 1 α; Gar,t 2 α; Gar,t 1 α; Gar,t 2 α; Gar,t 1 α; Gar,t 2 α; Gar,t 1 Gar,t 2 Singularity pattern 3 2 + 1 + 1 + 1 The Riemann scheme is given by and the Fuchs-Hukuhara relation is written as Here The Hamiltonians are given by Singularity pattern 5 2 + 1 + 1 The Riemann scheme is given by and the Fuchs-Hukuhara relation is written as θ t 1 + θ t 2 + θ ∞ 1 = 0. The Lax pair is given as Here The Hamiltonians are written as Singularity pattern 2 + 3 2 + 1 The Riemann scheme is given by and the Fuchs-Hukuhara relation is written as θ 0 + θ t 1 + θ ∞ 1 = 0. The Lax pair is expressed as Here A The Hamiltonians are given by .
The Riemann scheme is given by and the Fuchs-Hukuhara relation is written as θ 0 + θ ∞ 1 = 0. The Lax pair is expressed as Here The Hamiltonians are given by Gar,t 1 θ 0 ; Gar,t 2 θ 0 ; The Riemann scheme is given by and the Fuchs-Hukuhara relation is written as θ ∞ 1 + θ ∞ 2 = 0. The Lax pair is expressed as Here The Hamiltonians are given by Gar,t 1 2θ ∞ 1 ; Gar,t 2 2θ ∞ 1 ; The gauge parameter u satisfies Singularity pattern 5 2 + 2 The Riemann scheme is given by and the Fuchs-Hukuhara relation is written as θ 0 + θ ∞ 1 = 0. The Lax pair is expressed as The Hamiltonians are given by Gar,t 1 θ ∞ 1 ; Gar,t 2 θ ∞ 1 ; Singularity pattern 3 2 + 3 2 + 1 The Riemann scheme is given by and the Fuchs-Hukuhara relation is written as θ t 1 + θ ∞ 1 = 0. The Lax pair is expressed as The Hamiltonians are given by

Singularity pattern
The Riemann scheme is given by The Lax pair is expressed as Four-Dimensional Painlevé-Type Equations Associated with Ramified Linear Equations III 23 The Hamiltonians are given by

Degenerate Fuji-Suzuki systems
The Fuji-Suzuki systems were discovered by Fuji and Suzuki [3] in their study on similarity reductions of the Drinfeld-Sokolov hierarchies. A family of Painlevé-type equations which includes the Fuji-Suzuki system with A 5 -symmetry was independently proposed by Tsuda [31]. Sakai [28] derived the Fuji-Suzuki system of type A 5 from the isomonodromic deformation of the following Fuchsian system: where A 0 , A 1 , and A t are 3 × 3 matrices satisfying the following conditions and Thus the spectral type of the Fuchsian system (3.2) is 21, 21, 111, 111. Taking the trace of (3.3), we have the Fuchs relation The isomonodromic deformation equation of (3.2) is equivalent to the Hamiltonian system where the Hamiltonian is given by The linear system of the spectral type 21, 21, 111, 111 has one deformation parameter. However, linear systems which are degenerated from the 21, 21, 111, 111-system sometimes admit two-dimensional deformation, since a degeneration process does not necessarily preserve the number of deformation parameters. Such degenerations were pointed out in [15].

Laplace transform
In the degeneration scheme of the Garnier system and the Fuji-Suzuki system, we can see that the same Hamiltonian appears in several places. The linear systems corresponding to the same Hamiltonian can be transformed into one another by the Laplace transform. In this section, we present the correspondences through the Laplace transform. It is known that a linear system (2.1) with r ∞ ≤ 1 can canonically be written in the following form [34]: by the Laplace transform x → −d/dx, d/dx → x. The correspondence between (4.1) and (4.2) is known as the Harnad duality when both T and S are semisimple [6], while here we do not impose the semisimplicity of T or S. Using this, we have the following correspondences of linear systems: Here we indicated which spectral types correspond to ∞. When r ∞ is greater than one, the correspondence through the Laplace transform is somewhat complicated. However, by using the method described in [15], we obtain the following correspondences:  1))))((((1)))) .

Conclusion
The degeneration scheme presented in this series of papers focus on linear equations. Thus the scheme is redundant in terms of Hamiltonians since it happens that a certain Hamiltonian appears in several places. When we focus on Hamiltonians, the degeneration scheme can be reduced to the following scheme. This scheme shows the relationship among 40 Painlevé-type equations including those already known.
Remark 5.1. To determine whether the Hamiltonians given different names in this series of papers are actually different or not requires further consideration. Concerning this problem, we refer to [19], which is an attempt to characterize the four-dimensional Painlevé-type equations from an algebro-geometric point of view.
Remark 5.2. We think that there are no degenerations of linear systems other than what we considered in this series of papers, and hence we believe that there are no other four-dimensional Painlevé-type equations. However, further research is needed to show that the four-dimensional Painlevé-type equations obtained in this series of papers actually constitute a complete list. We think that a way to classify the unramified linear equations with four accessory parameters gives a hint (see [7]).

A.2 degenerations of the Fuji-Suzuki system
We first note that, concerning the following two linear systems, we adopt slightly different parametrizations from those in [15] for convenience of calculation. As for the linear system of the spectral type (11)(1), 21, 111, the irregular singular point is moved to x = ∞, so that the Riemann scheme is given by , and the Fuchs-Hukuhara relation is written as θ 0 1 + θ 0 2 + θ 1 + θ ∞ 1 + θ ∞ 2 + θ ∞ 3 = 0. The Lax pair is expressed as