Polynomial Hamiltonian system in two variables with $W({A}^{(1)}_1)$-symmetry and the second Painlevé hierarchy

On some Hamiltonian structures of coupled Painlevé II systems in dimension four

We find and study a two-parameter family of coupled Painlevé II systems in dimension four with affine Weyl group symmetry of several types. Moreover, we find a three-parameter family of polynomial Hamiltonian systems in two variables $t,s$. Setting $s=0$, we can obtain an autonomous version of the coupled Painlevé II systems. We also show its symmetry and holomorphy conditions.

Studies on the second member of the second Painlevé hierarchy

In this paper, we study the second member of the second Painlevé hierarchy $P_{II}^{(2)}$. We show that the birational transformations take this equation to the polynomial Hamiltonian system in dimension four, and this Hamiltonian system can be considered as a 1-parameter family of coupled Painlevé systems. This Hamiltonian is new. We also show that this system admits extended affine Weyl group symmetry of type $A_1^{(1)}$, and can be recovered by its holomorphy conditions. We also study a fifth-order ordinary differential equation satisfied by this Hamiltonian. After we transform this equatio…

Symmetry and holomorphy of the third-order ordinary differential equation satisfied by the third Painlevé Hamiltonian

We study symmetry and holomorphy of the third-order ordinary differential equation satisfied by the third Painlevé Hamiltonian.

准庞加莱型哈密顿系统的几何方法

Geometric approach for the identification of Hamiltonian systems of quasi-Painlevé type

本文通过几何方法识别准庞加莱型哈密顿系统，利用双有理变换分析其全局结构。

Hamiltonian system for the elliptic form of Painlevé VI equation

In literature, it is known that any solution of Painlevé VI equation governs the isomonodromic deformation of a second order linear Fuchsian ODE on $\mathbb{CP}^{1}$. In this paper, we extend this isomonodromy theory on $\mathbb{CP}^{1}$ to the moduli space of elliptic curves by studying the isomonodromic deformation of the generalized Lamé equation. Among other things, we prove that the isomonodromic equation is a new Hamiltonian system, which is equivalent to the elliptic form of Painlevé VI equation for generic parameters. For Painlevé VI equation with some special parameters, the isomonodr…

Cubic Pencils and Painlevé Hamiltonians

We present a simple heuristic method to derive the Painlevé differential equations from the corresponding geometry of rational surafces. We also give a direct relationship between the cubic pencils and Seiberg-Witten curves.