Dynamical systems and $\sigma$-symmetries

A deformation of the standard prolongation operation, defined on sets of vector fields in involution rather than on single ones, was recently introduced and christened ‘σ-prolongation’; correspondingly, one has ‘σ-symmetries’ of differential equations. These can be used to reduce the equations under study, but the general reduction procedure under σ-symmetries fails for equations of order 1. In this paper, we discuss how σ-symmetries can be used to reduce dynamical systems, i.e. sets of first-order ODEs in the form x'=f(x).