λ-Symmetries, isochronicity, and integrating factors of nonlinear ordinary differential equations

We obtain λ-symmetries of some second-order equations of the Painlevé-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 λ-symmetries of certain special types of third-order ODEs. Finally, we indicate a possible connection between λ-symmetries and the property of isochronicity for the Liénard equation of the second type.