Painlevé Vi Systems in Dimension Four with Affine Weyl Group Symmetry of Types

We find and study four kinds of 6-parameter family of coupled Painlevé VI systems with affine Weyl group symmetry of types B (1) 6 , D (1) 6 and D (2) 7 . We also give an explicit description of a confluence to the Noumi-Yamada system of type A (1) 5 . 0. Introduction In 1912, considering the significant problem of searching for higher order analogues of the Painlevé equations, Garnier discovered a series of systems of nonlinear partial differential equations, which can be considered as a generalization of the Painlevé VI equations from the viewpoint of monodromy preserving deformations of secondorder linear ordinary differential equations, now called the Garnier system (see [2]). In 1998, from the viewpoint of affine Weyl groups, Noumi and Yamada proposed a series of systems of nonlinear ordinary differential equations with affine Weyl group symmetry of type A (1) l (see [4, 5]). This series gives a generalization of Painlevé equations PIV and PV to higher orders. Furthermore, they clarified the relation between the Noumi-Yamada system and Soliton equations (a clear and elegant exposition on this can be found in [7]). The Painlevé VI equations have symmetry under the affine Weyl group of type D (1) 4 . On the other hand, the generalizations by Noumi and Yamada do not include the Painlevé VI equations. Thus, it is an important remaining problem to find a generalization of the Painlevé VI equations for which the symmetries can be established. In the present paper, we propose a 6-parameter family of four-dimensional coupled Painlevé VI systems with affine Weyl group symmetry of type D (1) 6 . Our differential system is equivalent to a Hamiltonian system given by dx dt = ∂H ∂y , dy dt = − ∂H ∂x , dz dt = ∂H ∂w , dw dt = − ∂H ∂z (1) with the Hamiltonian H = HV I(x, y, t;α0, α1, α2, α3, α4) +HV I(z, w, t; β0, β1, β2, β3, β4) + 2(x− t)yz{(z − 1)w + β2} t(t− 1) . (2) Here x, y, z, w denote unknown complex variables, and α0, α1, α2, α4, β2, β3, β4 are complex parameters satisfying the relation: α0 + α1 + 2α2 + 2(α4 − β4) + 2β2 + β3 + β4 = 1. (3) 2000Mathematics Subject Classification Numbers. 34M55, 34M45, 58F05, 32S65, 14E05, 20F55. 1