U. Ben-Ya'acov, "Localization of the relativistic centre-of-mass from internal dynamics", J. Phys. A: Math. Theor. 42 (2009) 225401.

he localization of the centre-of-mass (CM) of a two-body electromagnetic/gravitational system in the post-Newtonian approximation is shown to depend, in general, on the internal configuration of the system via the corresponding Newtonian Runge-Lenz vector. The requirement that the centre-of-mass shouldn't depend on the internal configuration then uniquely determined the CM cordinate in terms of the Runge-Lenz vector. A similar result is found for fully relativistic non-interacting particles. We conclude that consideration of the centre-of-mass of relativistic systems should involve the Runge-Lenz symmetry explicitly, as an essential part of the internal symmetries of the system.
http://stacks.iop.org/JPhysA/42/225401

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