Symmetry in the Painlev\'e systems and their extensions to four-dimensional systems

TL;DR

This paper introduces a new approach to understanding the symmetries of classical Painlevé equations and extends these symmetries to novel four-dimensional analogues, preserving their intrinsic symmetry properties.

Contribution

The paper presents a novel method for analyzing Painlevé symmetries and extends these to higher-dimensional systems while maintaining their symmetry structures.

Findings

New symmetry approach for Painlevé equations

Natural extensions to fourth-order analogues

Preservation of symmetries in extended systems

Abstract

We give a new approach to the symmetries of the Painlev\'e equations $P_{V},P_{IV},P_{III}$ and $P_{II}$, respectively. Moreover, we make natural extensions to fourth-order analogues for each of the Painlev\'e equations $P_{V}$ and $P_{III}$, respectively, which are natural in the sense that they preserve the symmetries.