Part 1. Isomonodromic deformation and 4-dimensional Painlevé-type equations

Abstract

The Painlevé equations are classified into eight types, and all of them are obtained from degenerations of the sixth Painlevé equation. This is the case of a two-dimensional phase space. When we consider the four-dimensional case, we have four systems that are sources, playing the same role that the sixth Painlevé equation does for the two-dimensional case. In this paper, we write down the Hamiltonian equations corresponding to all of the four systems that are obtained from the deformation theory of the Fuchsian equations. These are the well-known Garnier system with two independent variables, a Fuji-Suzuki system, a Sasano system, and the last one, which is new.