Lie point symmetries, partial Noether operators and first integrals of the Painlevé–Gambier equations

We utilize the Lie–Tressé linearization method to obtain linearizing point transformations of certain autonomous nonlinear second-order ordinary differential equations contained in the Painlevé–Gambier classification. These point transformations are constructed using the Lie point symmetry generators admitted by the underlying Painlevé–Gambier equations. It is also shown that those Painlevé–Gambier equations which have a few Lie point symmetries and hence are not linearizable by this method can be integrated by a quadrature. Moreover, by making use of the partial Lagrangian approach we obtain time dependent and time independent first integrals for these Painlevé–Gambier equations which have not been reported in the earlier literature. A comparison of the results obtained in this paper is made with the ones obtained using the generalized Sundman linearization method.