$$\lambda $$ -Symmetries, isochronicity, and integrating factors of nonlinear ordinary differential equations

Abstract

We obtain \(\lambda \)-symmetries of some second-order equations of the Painleve–Gambier type and study their relationship with the standard adjoint symmetry equation used for determining the integrating factor of a second-order ordinary differential equation (ODE). This is followed by a brief study of the \(\lambda \)-symmetries of certain special types of third-order ODEs. Finally, we indicate a possible connection between \(\lambda \)-symmetries and the property of isochronicity for the Lienard equation of the second type.