Solvable Structures Associated to the Nonsolvable Symmetry Algebra $\mathfrak{sl}(2,\mathbb{R})$

TL;DR

This paper demonstrates that third-order ODEs with nonsolvable symmetry algebra sl(2,R) admit solvable structures, enabling their integration by quadratures, thus extending solvability methods to certain nonsolvable symmetry cases.

Contribution

It introduces a method to construct solvable structures from sl(2,R) symmetry algebras, allowing integration of related differential equations.

Findings

Third-order ODEs with sl(2,R) symmetry admit solvable structures.

Such equations can be integrated by quadratures using these structures.

The approach generalizes solvability techniques to nonsolvable symmetry algebras.

Abstract

Third-order ordinary differential equations with Lie symmetry algebras isomorphic to the nonsolvable algebra $\mathfrak{sl}(2,\mathbb{R})$ admit solvable structures. These solvable structures can be constructed by using the basis elements of these algebras. Once the solvable structures are known, the given equation can be integrated by quadratures as in the case of solvable symmetry algebras.