### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 077, 18 pages      arXiv:1511.03025      https://doi.org/10.3842/SIGMA.2016.077

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

Department of Mathematics, University of Cádiz, 11510 Puerto Real, Spain

Received November 11, 2015, in final form August 03, 2016; Published online August 08, 2016

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.

Key words: first integral; solvable structure; $\mathcal{C}^{\infty}$-symmetry; nonsolvable algebra.

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