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

TL;DR

This paper introduces a polynomial Hamiltonian system with specific symmetry properties, links it to the second Painlevé hierarchy, and explores its relation to mKdV hierarchies, expanding understanding of integrable systems.

Contribution

It presents a new polynomial Hamiltonian system with $W({A}^{(1)}_1)$-symmetry derived from linear differential equations and connects it to Painlevé and mKdV hierarchies.

Findings

System with $W({A}^{(1)}_1)$-symmetry found

Relation established with second Painlevé hierarchy

Connections made to mKdV hierarchies

Abstract

We find a one-parameter family of polynomial Hamiltonian system in two variables with $W({A}^{(1)}_1)$-symmetry. We also show that this system can be obtained by the compatibility conditions for the linear differential equations in three variables. We give a relation between it and the second member of the second Painlev\'e hierarchy. Moreover, we give some relations between an autonomous version of its polynomial Hamiltonian system in two variables and the mKdV hierarchies.