L}{\partial\dot{\eta}}=\left(\xi^{2}+\eta^{2}\right)\dot{\eta},\label{eq:p_eta}\\
p_{\phi} & =\frac{\partial L}{\partial\dot{\phi}}=\xi^{2}\eta^{2}\dot{\phi},\label{eq:p_phi}\end{aligned}$$ and the Hamiltonian is written as $$\begin{aligned}
\mathcal{H} & =\dot{\xi}p_{\xi}+\dot{\eta}p_{\eta}+\dot{\phi}p_{\phi}-L\\
 & =\frac{1}{2}\frac{p_{\xi}^{2}+p_{\eta}^{2}}{\xi^{2}+\eta^{2}}+\frac{1}{2}\frac{p_{\phi}^{2}}{\xi^{2}\eta^{2}}-\frac{2\mu}{\xi^{2}+\eta^{2}}-\varepsilon\frac{\xi^{2}-\eta^{2}}{2}.
\label{eq:Ham_def}\end{aligned}$$ Since the coordinate $\phi$ is absent from the Hamiltonian, the momentum $p_{\phi}$ is a constant of motion. It can be checked by substitution that $p_{\phi}$ is the $z$ component of the total angular momentum of the system. Thus, when $p_{\phi}$ vanishes, the motion is confined to a plane perpendicular to the $xy$ plane and intersecting the origin, and we can refer to this subcase as the *bidimensional* problem (as opposed to the *three-dimensional* problem when $p_{\phi}$ is not null).

We now employ a Sundman regularisation [@sundman_memoire_1912], introducing the fictitious time $\tau$ via the differential relation $$dt=\left(\xi^{2}+\eta^{2}\right)d\tau,\label{eq:fic_time}$$ and the new, identically null, function $$\mathcal{H}_{\tau}\left(p_\xi,p_\eta,p_\phi;\xi,\eta,\phi\right)=\left(\mathcal{H}-h\right)\left(\xi^{2}+\eta^{2}\right),$$ where $h$ is the energy constant of the system (obtained by substituting the initial conditions into the expression of $\mathcal{H}$). We have then for $p_{\xi}$ and $\xi$ $$\begin{aligned}
\frac{dp_{\xi}}{d\tau} & =\frac{dp_{\xi}}{dt}\frac{dt}{d\tau}=-\frac{\partial\mathcal{H}}{\partial\xi}\left(\xi^{2}+\eta^{2}\right)=-\frac{\partial\mathcal{H}_{\tau}}{\partial\xi},\\
\frac{d\xi}{d\tau} & =\frac{d\xi}{dt}\frac{dt}{d\tau}=\frac{\partial\mathcal{H}}{\partial p_{\xi}}\left(\xi^{2}+\eta^{2}\right)=\frac{\partial\mathcal{H}_{\tau}}{\partial p_{\xi}},\end{aligned}$$ and, similarly for $p_{\eta}$, $\eta$, $p_{\phi}$ and $\phi$, $$\begin{aligned}
\frac{dp_{\eta}}{d\tau} & = -\frac{\partial\mathcal{H}_{\tau}}{\partial\eta},\\
\frac{d\eta}{d\tau} & = \frac{\partial\mathcal{H}_{\tau}}{\partial p_{\eta}},\\
\frac{dp_{\phi}}{d\tau} & = -\frac{\partial\mathcal{H}_{\tau}}{\partial\phi},\\
\frac{d\phi}{d\tau} & = \frac{\partial\mathcal{H}_{\tau}}{\partial p_{\phi}}.\end{aligned}$$ $\mathcal{H}_\tau$ can thus be considered as an Hamiltonian function describing the evolution of the system in 