摘要翻译：
本文研究了第二个Painlev族$P_{II}^{（2）}$的第二个成员。我们证明了双分变换将该方程转化为维数为4的多项式哈密顿系统，并且该哈密顿系统可以看作是耦合Painlev系统的1-参数族。这个哈密顿量是新的。我们还证明了该系统具有$A_1^{（1）}$型的扩展仿射Weyl群对称性，并可由其全纯条件恢复。我们还研究了这个哈密顿量所满足的一个五阶常微分方程。通过二次变换将该方程转化为五维多项式型一阶常微分方程组，给出了它的对称性和全纯性条件。
---
英文标题：
《Studies on the second member of the second Painlev\'e hierarchy》
---
作者：
Yusuke Sasano
---
最新提交年份：
2009
---
分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
--
一级分类：Mathematics        数学
二级分类：Classical Analysis and ODEs        经典分析与颂歌
分类描述：Special functions, orthogonal polynomials, harmonic analysis, ODE's, differential relations, calculus of variations, approximations, expansions, asymptotics
特殊函数、正交多项式、调和分析、Ode、微分关系、变分法、逼近、展开、渐近
--

---
英文摘要：
  In this paper, we study the second member of the second Painlev\'e 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 Painlev\'e 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 equation into a system of the first-order ordinary differential equations of polynomial type in dimension five by birational transformations, we give its symmetry and holomorphy conditions.
---
PDF链接：
https://arxiv.org/pdf/0805.2444