Symmetry Lie algebras of nth order ordinary differential equations

Abstract

We show that an nth ( n ⩾ 3) order linear ordinary differential equation has exactly one of n + 1, n + 2, or n + 4 (the maximum) point symmetries. The Lie algebras corresponding to the respective numbers of point symmetries are obtained. Then it is shown that a necessary and sufficient conditon for an nth ( n ⩾ 3) order equation to be linearizable via a point transformation is that it must admit the n dimensional Abelian algebra nA 1 = A 1 ⊕ A 1 ⊕ … ⊕ A 1. We discuss in detail the symmetry realizations of ( n − 1) A 1 ⊕ s A 1. Finally, we prove that an nth ( n ⩾ 3) order equation q ( n) = H( t, q, …, q n − 1 ) cannot admit exactly an n + 3 dimensional algebra of point symmetries which is a subalgebra of nA 1 ⊕, gl(2, R ).