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

We give a reformulation of a six-parameter family of coupled Painlevé VI systems with affine Weyl group symmetry of type D (1) 6 from the viewpoint of its symmetry and holomorphy properties. In [9, 10], we proposed a 6-parameter family of four-dimensional coupled Painlevé VI systems with affine Weyl group symmetry of type D (1) 6. This system can be considered as a genelarization of the Painlevé VI system. In this paper, from the viewpoint of its symmetry and holomorphy properties we give a reformulation of this system (cf. [11]) explicitly given by dq 1 dt = ∂H ∂p 1 , dp 1 dt = − ∂H ∂q 1 , dq 2 dt = ∂H ∂p 2 , dp 2 dt = − ∂H ∂q 2 , + 2(q 1 − η)q 2 {(q 1 − t)p 1 + α 2 }{(q 2 − 1)p 2 + α 4 } t(t − 1)(t − η) (η ∈ C − {0, 1}).