A note on generalization of the Lewis invariant and the Ermakov systems

In this paper we examine several possible generalizations of the well known Lewis invariant and the Ermakov systems. The analysis relies on the applic ation of the theorem of Emmy Noether. The problem of constructing Lewis-typ e invariants is reduced to a problem of solving a single partial differenti al equation-the so-called potential equation. Within this approach, the rol e of the auxiliary equation is clarified by an analysis of linear systems. Generalization of the Ermakov-type procedure for derivation of the Lewis in variant is also discussed. Moreover, different forms of general solution of the problem will be provided, which extend the range of possible applicati ons of the analysis.