The classical Taub-Nut System: factorization, spectrum generating algebra and solution to the equations of motion

The formalism of SUSYQM (SUperSYmmetric Quantum Mechanics) is properly modified in such a way to be suitable for the description and the solution of a classical maximally superintegrable Hamiltonian System, the so-called Taub-Nut system, associated with the Hamiltonian: $$\mathcal{H}_\eta ({\mathbf{q}}, {\mathbf{p}}) = \mathcal{T}_\eta ({\mathbf{q}}, {\mathbf{p}}) + \mathcal{U}_\eta({\mathbf{q}}) = \frac{|{\mathbf{q}}| {\mathbf{p}}^2}{2m(\eta + |{\mathbf{q}}|)} - \frac{k}{\eta + |{\mathbf{q}}|} \quad (k>0, \eta>0) \, .$$ In full agreement with the results recently derived by A. Ballesteros et al. for the quantum case, we show that the classical Taub-Nut system shares a number of essential features with the Kepler system, that is just its Euclidean version arising in the limit $\eta \to 0$, and for which a SUSYQM approach has been recently introduced by S. Kuru and J. Negro. In particular, for positive $\eta$ and negative energy the motion is always periodic; it turns out that the period depends upon $\eta$ and goes to the Euclidean value as $\eta \to 0$. Moreover, the maximal superintegrability is preserved by the $\eta$-deformation, due to the existence of a larger symmetry group related to an $\eta$-deformed Runge-Lenz vector, which ensures that in $\mathbb{R}^3$ closed orbits are again ellipses. In this context, a deformed version of the third Kepler's law is also recovered. The closing section is devoted to a discussion of the $\eta<0$ case, where new and partly unexpected features arise.