m at h . A G ] 2 1 Ju n 20 07 COUPLED PAINLEVÉ VI SYSTEMS IN DIMENSION FOUR WITH AFFINE WEYL GROUP SYMMETRY OF TYPE

We give a reformulation of a six-parameter family of coupled Painlevé VI systems with affine Weyl group symmetry of type D (1) 6 . We also study some Hamiltonian structures of this system. 0. Introduction 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 the Hamiltonian system given by dq1 dt = ∂H ∂p1 , dp1 dt = − ∂H ∂q1 , dq2 dt = ∂H ∂p2 , dp2 dt = − ∂H ∂q2 , H =HV I(q1, p1, η, t;α0, α1, α2, α3 + 2α4 + α5, α3 + α6) +HV I(q2, p2, η, t;α0 + 2α2 + α3, α1 + α3, α4, α5, α6) + 2(q1 − η)q2{(q1 − t)p1 + α2}{(q2 − 1)p2 + α4} t(t− 1)(t− η) (η ∈ C − {0, 1}). (1) Here q1, p1, q2, p2 denote unknown complex variables, and α0, α1, . . . , α6 are complex parameters satisfying the relation α0 + α1 + 2(α2 + α3 + α4) + α5 + α6 = 1, where the symbol HV I(q, p, η, t; β0, β1, β2, β3, β4) is given in Section 2. If we take the limit η → ∞, we obtain the Hamiltonian system with polynomial Hamiltonian H̃ H̃ = H̃V I(q1, p1, t;α0, α1, α2, α3 + 2α4 + α5, α3 + α6) + H̃V I(q2, p2, t;α0 + α3, α1 + 2α2 + α3, α4, α5, α6) + 2(q1 − t)p1q2{(q2 − 1)p2 + α4} t(t− 1) (2) given in [4], where the symbol H̃V I denotes H̃V I(q, p, t; δ0, δ1, δ2, δ3, δ4) = 1 t(t− 1) [p(q − t)(q − 1)q − {(δ0 − 1)(q − 1)q + δ3(q − t)q + δ4(q − t)(q − 1)}p+ δ2(δ1 + δ2)q] (δ0 + δ1 + 2δ2 + δ3 + δ4 = 1). (3) We can obtain the system (1) by the following steps: (1) We symmetrize the holomorphy conditions r i of the system (2) (see Section 5). 2000Mathematics Subject Classification Numbers. 34M55, 34M45, 58F05, 32S65, 14E05, 20F55. 1