On the reduction methods for ordinary differential equations

As tandard method based on the use of differential invariants of a Lie group, G ,e nables us to reduce any ordinary differential equation invariant under the action of G .W e show that this method is applicable to vector fields more general than those associated with Lie symmetries. We characterize all such vector fields and study their relationship with nonlocal symmetries and λsymmetries (Govinder K S and Leach P G L 1995 J. Phys. A: Math. Gen. 28 5349–59, Muriel C and Romero L 2001 IMA J. Appl. Math. 66 111–25).