Lie-Point Symmetries Preserved by Derivative

Abstract

Conditions to guarantee that a point symmetry ${\bf X}$ of an $n^{th}$-order differential equation $q^{(n)}-\omega=0$ is simultaneously a point symmetry of its derived equation $q^{(n+1)}-\dot{\omega}=0$ are analyzed, and the possible types of vector fields established. It is further shown that only the simple Lie algebra $\mathfrak{sl}(2,\mathbb{R})$ for a very specific type of realization in the plane can be inherited by a derived equation.