  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.
