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.
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