Some physical applications of systems of differential equations admitting a superposition rule

We give simple examples of applications in physics of the result of the theorem by Lie and Scheffers concerning systems of differential equations admitting a superposition function allowing us to write the general solution in terms of a set of arbitrary, but independent, particular solutions. The main idea consists in associating the equation on the given carrier space with an equation on a Lie group G of transformations of a manifold M. The universal character of these equations on the group G, which may arise either from classical systems or from quantum dynamical systems, will be illustrated with several examples of physical relevance, and we will also prove the usefulness of a generalization of a method proposed by Wei and Norman for solving these systems.