Initial value space of the four dimensional Painlev\'{e} system with $(A_5+A_1)^{(1)}$ symmetry

TL;DR

This paper constructs the initial value space for a four-dimensional Painlevé system with specific affine Weyl group symmetry, extending the geometric understanding of these integrable systems.

Contribution

It introduces the initial value space for a new four-dimensional Painlevé system with $(A_5+A_1)^{(1)}$ symmetry, expanding the geometric framework of Painlevé equations.

Findings

Constructed the initial value space as a symplectic manifold.

Described the Painlevé system as a polynomial Hamiltonian system.

Extended geometric analysis to a higher-dimensional Painlevé system.

Abstract

The initial value spaces of the Painlev\'{e} equations are proposed by Okamoto. They are symplectic manifolds in which the Painlev\'{e} equations are described as polynomial Hamiltonian systems on all coordinates. In this article, we construct an initial value space of the four dimensional Painlev\'{e} system with affine Weyl group symmetry of type $(A_5+A_1)^{(1)}$.