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A general procedure for understanding Kepler-type dynamical symmetries is presented. The main concern is the geodesic motion in Euclidean Taub–NUT space, which approximates the scattering of self-dual monopoles for long distances. Other examples include a test particle moving in the asymptotic field of a self-dual monopole and two other related metrics.

The manifold M of null rays through the origin of R 2 ' π+1 is diffeomorphic to S 1 x S", and it is a homogeneous space of SO(2,n + 1). This group therefore acts on T*M, which we show to be the "generating manifold" of the extended phase space of the regularized Kepler Problem. A local canonical chart in T*M is found such that the restriction to the subbundle of the null nonvanishing covectors is given by p 0 + H(q, p) = 0, where H(q, p) is the Hamiltonian of the Kepler Problem. By means of this construction, we get some results that clarify and complete the previous approaches to the problem.

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