NONLOCAL SYMMETRIES OF FIRST-ORDER EQUATIONS

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