Applications of $$\mathcal{C}^{\infty }$$ -Symmetries in the Construction of Solvable Structures

Abstract

A complete set of first integrals for a third order ordinary differential equation (ODE) that admits the non-solvable symmetry algebra \(\mathfrak{s}\mathfrak{l}(2, \mathbb{R})\) can be found by quadratures. These first integrals arise from a solvable structure that can be constructed in terms of two first integrals associated to \(\mathcal{C}^{\infty }\)-symmetries of a reduced second order ODE. The general procedure is illustrated by an explicit example where three independent first integrals of the third order equation are provided in terms of a complete set of solutions to a second order linear ODE.