We introduce a new class of symmetries, that strictly includes Lie symmetries, for which there exists an algorithm that lets us reduce the order of an ordinary differential equation. Many of the known order-reduction processes, that are not consequence of the existence of Lie symmetries, are a consequence of the invariance of the equation under vector fields of the new class. These vector fields must satisfy a new prolongation formula and there must exist a procedure for determining the vector fields of this class that lead to an equation invariant. We have also found some whose Lie symmetries are trivial, have no obvious order reductions, but can be completely integrated by using the new class of symmetries.

If an ordinary differential equation admits the non‐solvable Lie algebra sl(2, R) as symmetry algebra then, when the classical Lie method of reduction is applied, at least one of its generators cannot be used to obtain a second order reduction. In this paper it is proved that these generators can be recovered as C^∞‐symmetries of the reduced equations.TheseC^∞‐symmetries can be used to new‐order reductions if the order of the last reduced equations is higher than one. As a consequence, a classification of the third‐order equations that admit sl(2, R) as symmetry algebra is given. This step by step method of reduction is applied to the Chazy equation.