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

TL;DR

This paper introduces -symmetries, a deformation of the prolongation operation for vector fields, enabling a fully algorithmic reduction of ODEs with suitable invariance properties, extending standard symmetry methods.

Contribution

It generalizes -symmetries as a deformation of prolongation, maintaining invariants by differentiation, and allows systematic reduction of ODEs beyond classical symmetries.

Findings

-symmetries preserve invariants by differentiation.

The method enables algorithmic reduction of ODEs.

It extends the applicability of symmetry-based reduction techniques.

Abstract

We consider a deformation of the prolongation operation, defined on sets of vector fields and involving a mutual interaction in the definition of prolonged ones. This maintains the "invariants by differentiation" property, and can hence be used to reduce ODEs satisfying suitable invariance conditions in a fully algorithmic way, similarly to what happens for standard prolongations and symmetries.