On the generation of classes of planar systems with given orbital symmetries

Following the procedure used by Kamke, on the basis of the concepts of symmetry and orbital symmetry introduced by S. Lie, aim of this paper is to extend such a reasoning to classify systems for which the semi-invariants can be computed in closed-form, whence possibly used for the computation of Lyapunov functions as well. The concept of semi-invariant extends both the concept of first integral (a constant of motion) and the concept of eigenfunction, whence it seems to be the natural tool for the construction of Lyapunov functions for nonlinear systems.