Non-local symmetries of first-order equations

Abstract

It is shown how one can transform scalar first-order ordinary differential equations which admit non-local symmetries of the exponential type to integrable equations admitting canonical exponential non-local symmetries. As examples we invoke the Abel equation of the second kind, the Riccati equation and natural generalizations of these. Moreover, our method describes how a double reduction of order for a second-order ordinary differential equation which admits a two-dimensional Lie algebra of generators of point symmetries can be effected if the second-order equation is first reduced in order once by a symmetry which does not span an ideal of the two-dimensional Lie algebra.