Characteristic Lie rings and symmetries of differential Painlevé I and Painlevé III equations

Abstract

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

The article has been significantly revised, submitted to SIGMA