Nonlocal Symmetries, Telescopic Vector Fields and $\lambda$-Symmetries of Ordinary Differential Equations

TL;DR

This paper explores the connections between nonlocal symmetries, telescopic vector fields, and $\lambda$-symmetries in ordinary differential equations, offering new insights into order reduction methods and their computational advantages.

Contribution

It establishes relationships between various symmetry approaches and introduces the concept of equivalent $\lambda$-symmetries to distinguish different reductions.

Findings

$\lambda$-symmetries enable order reduction with fewer unknowns

Connections between nonlocal symmetries and $\lambda$-coverings are clarified

Equivalent $\lambda$-symmetries help identify genuinely different reductions

Abstract

This paper studies relationships between the order reductions of ordinary differential equations derived by the existence of $\lambda$-symmetries, telescopic vector fields and some nonlocal symmetries obtained by embedding the equation in an auxiliary system. The results let us connect such nonlocal symmetries with approaches that had been previously introduced: the exponential vector fields and the $\lambda$-coverings method. The $\lambda$-symmetry approach let us characterize the nonlocal symmetries that are useful to reduce the order and provides an alternative method of computation that involves less unknowns. The notion of equivalent $\lambda$-symmetries is used to decide whether or not reductions associated to two nonlocal symmetries are strictly different.