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

34A34Nonlinear ODE and systems, general
22E70Applications of Lie groups to physics; explicit representations
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