Generalized notions of symmetry of ODE's and reduction procedures

This paper describes the notion of \sigma -symmetry, which extends the one of \lambda-symmetry, and its application to reduction procedures of systems of ordinary differential equations and of dynamical systems as well. We also consider orbital symmetries, which give rise to a different form of reduction of dynamical systems. Finally, we discuss how dynamical systems can be transformed into higher-order ordinary differential equations, and how these symmetry properties of the dynamical systems can be transferred into reduction properties of the corresponding ordinary differential equations. Many examples illustrate the various situations.