U. Ben-Ya'acov, "Laplace–Runge–Lenz-like new constant in many-body systems from post-Newtonian dynamics", J. Phys. A: Math. Theor. 42 (2009) 375210

It has been shown in the past that the classical Laplace-Runge-Lenz vector emerges naturally in the computation of the post-Newtonian Lorentz boost in 2-body electrical or gravitational systems. This procedure is extended here to many-body systems. A new N-body vector observable of the Runge-Lenz type is found which is an integral of motion for nontrivial families of solutions. Conditions for its integrability are discussed with explicit examples. The relation of this vector with the post-Newtonian centre-of-mass is also briefly discussed.
http://stacks.iop.org//JPhysA/42/375210

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