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 nA1 = A1 ⊕ A1 ⊕ … ⊕ A1. We discuss in detail the symmetry realizations of (n − 1)A1 ⊕s A1. Finally, we prove that an nth (n ⩾ 3) order equation q(n) = H(t, q, …, qn − 1) cannot admit exactly an n + 3 dimensional algebra of point symmetries which is a subalgebra of nA1 ⊕, gl(2, R ).

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