POINT TRANSFORMATIONS AND PAINLEVÉ EQUATIONS

We consider the point-invariant classes of the ordinary differential equations. In particular, the point-invariant class of the second order ODE's that contains the Painlevé equations and the point-invariant class of the third order ODE's that contains the Chazy equations are distinguished. Tests for the equivalence of the second order ODE's to the Painlevé I and II equations are indicated. The necessary condition of the equivalence of the second order ODE's to the Painlevé IV and Painlevé III in the case a = 0 are shown.