### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 1 (2005), 018, 15 pages      nlin.SI/0511055      https://doi.org/10.3842/SIGMA.2005.018

### Ermakov's Superintegrable Toy and Nonlocal Symmetries

P.G.L. Leach a, A. Karasu (Kalkanli) b, M.C. Nucci c and K. Andriopoulos d
a) School of Mathematical Sciences, Howard College, University of KwaZulu-Natal, Durban 4041, Republic of South Africa
b) Department of Physics, Middle East Technical University, 06531 Ankara, Turkey
c) Dipartimento di Mathematica e Informatica, Università di Perugia, 06123 Perugia, Italy
d) Department of Information and Communication Systems Engineering, University of the Aegean, Karlovassi 83 200, Greece

Received September 19, 2005, in final form November 11, 2005; Published online November 15, 2005

Abstract
We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the algebra sl(2, R). The number of point symmetries is insufficient and the algebra unsuitable for the complete specification of the system. We use the method of reduction of order to reduce the nonlinear fourth-order system to a third-order system comprising a linear second-order equation and a conservation law. We obtain the representation of the complete symmetry group from this system. Four of the required symmetries are nonlocal and the algebra is the direct sum of a one-dimensional Abelian algebra with the semidirect sum of a two-dimensional solvable algebra with a two-dimensional Abelian algebra. The problem illustrates the difficulties which can arise in very elementary systems. Our treatment demonstrates the existence of possible routes to overcome these problems in a systematic fashion.

Key words: Ermakov system; reduction of order; complete symmetry group.

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