Generalized Solvable Structures and First Integrals for ODEs Admitting an sl(2,R) Symmetry Algebra

The notion of solvable structure is generalized in order to exploit the presence of an sl(2,R) algebra of symmetries for a kth-order ordinary differential equation E with k > 3. In this setting, the knowledge of a generalized solvable structure for E allows us to reduce E to a family of second-order linear ordinary differential equations depending on k − 3 parameters. Examples of explicit integration of fourth and fifth order equations are provided in order to illustrate the procedure.