Symmetries of differential equations

The use of a symmetry to reduce the order of an nth-order differential equation is treated by considering the symmetry of an associated vector field. A particular choice of associated vector field leads to the usual extension of the Lie symmetry method. The possibility of other choices leads to a powerful generalisation. An algebraic classification of transformations arises naturally from the theory. It is shown to be equivalent to the geometric classification of transformations as contact and non-contact.