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

Two-component hyperbolic system of equations generated by ordinary differential Painlev'e I $$u_{yy}=6u^2+y$$ and Painlev'e III $$yu{u}_{yy}=y{u}_y^2-u{u}_y+\delta y+\beta u+\alpha u^3+\gamma y{u}^4$$ 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.