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We investigate the symmetry properties of a pair of Ermakov equations. Thesystem is superintegrable and yet possesses only three Lie point symmetrieswith the algebra sl(2,R). The number of point symmetries is insufficient andthe algebra unsuitable for the complete specification of the system. We use themethod of reduction of order to reduce the nonlinear fourth-order system to athird-order system comprising a linear second-order equation and a conservationlaw. We obtain the representation of the complete symmetry group from thissystem. Four of the required symmetries are nonlocal and the algebra is thedirect sum of a one-dimensional Abelian algebra with the semidirect sum of atwo-dimensional solvable algebra with a two-dimensional Abelian algebra. Theproblem illustrates the difficulties which can arise in very elementarysystems. Our treatment demonstrates the existence of possible routes toovercome these problems in a systematic fashion.

Ermakov's Superintegrable Toy and Nonlocal Symmetries