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

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