On generalized Sundman transformation method, first integrals, symmetries and solutions of equations of Painlevé–Gambier type

Abstract

We employ the generalized Sundman transformation method to obtain certain new first integrals of autonomous second-order ordinary differential equations belonging to the Painlevé–Gambier classification scheme. This method not only yields systematically the known first integrals of a large number of the Painlevé–Gambier equations but also some time dependent ones, which greatly simplify the computation of their corresponding solution. In addition we also compute the Sundman symmetries of these equations.