Reductions for Some Ordinary Differential Equations Through Nonlocal Symmetries

In [19] we derive nonlocal symmetries for ordinary differential equations by embedding the given equation in an auxiliary system. Since the nonlocal symmetries of the ODE’s are local symmetries of the associated auxiliary system this result provides an algorithmic method to derive this kind of nonlocal symmetries. In this work we show some classes of ordinary differential equations which do not admit any Lie symmetry unless some conditions are satisfied but for which we have derived nonlocal symmetries. These nonlocal symmetries allow us to reduce the order for these equations even if these equations do not admit point symmetries.