  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.
