Using 1-1 mappings, the complete symmetry groups of contact transformations of general linear second-order ordinary differential equations are determined from two independent solutions of those equations, and applied to the harmonic oscillator with and without damping.
Martini, R., & Kersten, P. H. M. (1983). Contact symmetries of general linear second-order ordinary differential equations: letter to the editor. Journal of physics A: mathematical and general, 16(13), L455. https://doi.org/10.1088/0305-4470/16/13/003