arXiv:nlin/0107021v2 [nlin.SI] 2 Oct 2002 Journal of Nonlinear Mathematical Physics Volume 8, Number 4 (2001), 475–490 Article Symmetry, Singularities and Integrability in Complex Dynamics V: Complete Symmetry Groups of Certain Relativistic Spherically Symmetric Systems P G L LEACH †∗ , M C NUCCI ‡ and S COTSAKIS † † GEODYSYC, Department of Mathematics, University of the Aegean, Karlovassi 83 200, Greece ∗ Permanent address: School of Mathematical and Statistical Sciences, University of Natal, Durban 4041, Republic of South Africa ‡ Dipartimento di Matematica e Informatica, Universit` a di Perugia, 06123 Perugia, Italy Received August 3, 2000; First Revision December 7, 2000; Second Revision May 4, 2001; Accepted May 5, 2001 Abstract We show that the concept of complete symmetry group introduced by Krause (J. Math. Phys. 35 (1994), 5734–5748) in the context of the Newtonian Kepler problem has wider applicability, extending to the relativistic context of the Einstein equations de- scribing spherically symmetric bodies with certain conformal Killing symmetries. We also provide a simple demonstration of the nonuniqueness of the complete symmetry group. 1 Introduction In 1994 Krause [12] introduced the concept of a complete symmetry group in classical me- chanics. This concept was somewhat stronger than that of the standard symmetry group generally used at the time, which was the group represented by the Lie point symme- tries of the system of differential equations describing the dynamical system. A complete symmetry group realisation in mechanics was required to be endowed with the properties that (i) the group act freely and transitively on the manifold of all allowed motions of the system and (ii) the given equations of motion be the only ordinary differential equations that remain invariant under the specified action of the group. The vehicle which Krause used to demonstrate the concept was the classical Kepler problem described in reduced coordinates by the second order ordinary differential equation ¨ r + µr r 3 =0, (1.1) Copyright c  2001 by P G L Leach, M C Nucci and S Cotsakis