Integration of Ordinary Differential Equations via Nonlocal Symmetries

We present a nonlocal symmetry method to reduce scalar first- and second-orderordinary differential equations (ODEs) to quadratures. It is shown that a second-orderODE admitting a non-Abelian two-dimensional Lie algebra of point symmetriesis reducible to quadratures via a nonideal of the algebra. We also providea direct method of integration for a first-order ODE admitting an exponential nonlocal symmetry which satisfies an additional property.Moreover, we give examples, two on double reduction and several on Abel equations of the second kind, that illustrate ourapproaches.