Characteristic Lie rings and symmetries of differential Painlev\'e I and Painlev\'e III equations

TL;DR

This paper investigates the structure of characteristic Lie rings and higher symmetries of hyperbolic systems derived from Painlevé I and III equations, revealing new insights into their symmetry properties.

Contribution

It introduces the analysis of characteristic Lie rings and constructs higher Lie-Bäcklund symmetries for systems related to Painlevé I and III equations.

Findings

Structure of characteristic Lie rings is characterized.

Higher Lie-Bäcklund symmetries are explicitly constructed.

Results deepen understanding of symmetries in Painlevé-related systems.

Abstract

Two-component hyperbolic system of equations generated by ordinary differential Painlev\'e I \[ u_{yy}=6u^2+y \] and Painlev\'e III \[ yuu_{yy}=yu^2_{y}-uu_y+\delta y+\beta u+\alpha u^3 +\gamma yu^4 \] equations are considered, where $\alpha, \beta, \gamma, \delta$ are complex numbers. The structure of characteristic Lie rings is studied and higher symmetries of Lie-B\"acklund are obtained.