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The Lie algebra sl(3,$ℛ\right)$ and linearization. (English) Zbl 0683.34004
Summary: In a previous paper [J. Phys. A 20, 277-292 (1987; Zbl 0625.34044)] the authors and W. Sarlet established the form of second-order ordinary differential equations with two commuting symmetries (in canonical form ${G}_{1}=\partial /\partial t$, ${G}_{2}=\partial /\partial q$, ${G}_{2}\ne \rho \left(q,t\right){G}_{1}\right)$ which have the Lie algebra sl(3,$ℛ\right)$. In this paper, we determine the conditions under which an equation with two non- commuting (non-proportional) symmetries possesses the Lie algebra sl(3,$ℛ\right)$. We also obtain the most general nonlinear equation at most linear in the first derivative which has sl(3,$ℛ\right)$ algebra.
##### MSC:
 34A34 Nonlinear ODE and systems, general 22E70 Applications of Lie groups to physics; explicit representations
##### Keywords:
commuting symmetries