A generalization of \(\lambda\)-symmetry reduction for systems of ODEs: \(\sigma\)-symmetries (Q2914400)

Consider a scalar ODE of order \(n>1\) in \(J^nM\), the jet bundle of order \(n\) over the manifold \(M\) of independent and dependent variables. The present paper is devoted to the case when the given ODE is invariant under a set of vector fields defined on \(J^nM\), and obtained from vector fields in \(M\) under a further modified version of \(\lambda \)-prolongation. More precisely, in this work, the modified prolongation operation does not act on a single vector field but rather on the set of vector fields, that is, the prolongation of each of them involves the other ones. This joint-\(\lambda \) prolongation depends on a matrix \(\sigma \) defined by a set of smooth functions on \(J^1M\), and is therefore called \(\sigma \)-prolongation. The authors obtain that, even in this case, the standard approach to reduction is valid. Nine examples are completely treated.