Equivalence of second-order ordinary differential equations to Painlevé equations

Abstract

All Painleve equations except the first belong to one type of equations. In terms of invariants of these equations, we obtain criteria for the equivalence to the second Painleve equation and to equation XXXIV in the list of 50 equations without movable critical points. We find new necessary conditions of equivalence for the third and fourth and also special cases of the fifth and sixth Painleve equations. We compare the invariants we use with invariants previously introduced by other authors and compare the obtained results.