Unique distinction of the Lorentz and the anti-de Sitter group in the classical Kepler problem

In contrast to the familiar group-theoretical investigations of the Kepler problem, providing rather different symmetry structures for different types of state, the question is posed whether the problem does not exhibit uniquely distinguished dynamic invariance symmetries, which may be considered characteristic for the Newtonian interaction, independent of the special type of orbit. Taking into account only one Laplace vector it is shown that one, and only one, such symmetry exists and that this is just that of the Lorentz group. One is led to consider a distinguished orthonormal dynamical frame of the system, providing an extension of the Lorentz invariance symmetry to that of the anti-de Sitter group. Finally, some remarks are made concerning the relativistic formulation and solution of the Kepler problem. (author)