Symmetry Classification of Scalar $n$th Order Ordinary Differential Equations

Abstract: We complete the Lie symmetry classification of scalar nth order, $n \geq 4$, ordinary differential equations by means of the symmetry Lie algebras they admit. It is known that there are three types of such equations depending upon the symmetry algebra they possess, viz. first-order equations which admit infinite dimensional Lie algebra of point symmetries, second-order equations possessing the maximum eight point symmetries and higher-order, $n \geq 3$, admitting the maximum $n + 4$ dimensional symmetry algebra. We show that nth order equations for $n \geq 4$ do not admit maximally an $n + 3$ dimensional Lie algebra except for $n = 5$ which can admit $sl(3, R)$ algebra. Also, they can possess an $n + 2$ dimensional Lie algebra that gives rise to a nonlinear equation that is not linearizable via a point transformation. It is shown that for $n \geq 5$ there is only one such class of equations.