  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.
