$\lambda-$Symmetries and integrability by quadratures

TL;DR

This paper explores how $\\lambda$-symmetries can be used to solve second-order ordinary differential equations by quadratures, constructing commuting symmetries and deriving solutions through integrating factors and Jacobi last multipliers.

Contribution

It introduces a method utilizing two generalized $\lambda$-symmetries to solve second-order ODEs by quadratures, including explicit solutions for complex cases like Painlevé-Gambier.

Findings

Method constructs two commuting generalized symmetries.

Functions related to $\lambda$-symmetries yield integrating factors.

Explicit solutions obtained for a Painlevé-Gambier case.

Abstract

It is investigated how two (standard or generalized) $\lambda-$symmetries of a given second-order ordinary differential equation can be used to solve the equation by quadratures. The method is based on the construction of two commuting generalized symmetries for this equation by using both $\lambda-$symmetries. The functions used in that construction are related with integrating factors of the reduced and auxiliary equations associated to the $\lambda-$symmetries. These functions can also be used to derive a Jacobi last multiplier and two integrating factors for the given equation. Some examples illustrate the method; one of them is included in the XXVII case of the Painlev\'e-Gambier classification. An explicit expression of its general solution in terms of two fundamental sets of solutions for two related second-order linear equations is also obtained.