fictitious time[^2]. Explicitly, $$\begin{gathered}
\mathcal{H}_{\tau}=-\varepsilon\frac{\xi^{4}}{2}-h\xi^{2}+\frac{1}{2}p_{\xi}^{2}+\frac{1}{2}\frac{p_{\phi}^{2}}{\xi^{2}}\\
+\varepsilon\frac{\eta^{4}}{2}-h\eta^{2}+\frac{1}{2}p_{\eta}^{2}+\frac{1}{2}\frac{p_{\phi}^{2}}{\eta^{2}}-2\mu,\end{gathered}$$ and the Hamiltonian $\mathcal{H}_\tau$ has thus been separated into the two independent constants of motion $$\begin{aligned}
\alpha_{1} & =-\varepsilon\frac{\xi^{4}}{2}-h\xi^{2}+\frac{1}{2}p_{\xi}^{2}+\frac{1}{2}\frac{p_{\phi}^{2}}{\xi^{2}},\label{eq:alpha1}\\
\alpha_{2} & =\varepsilon\frac{\eta^{4}}{2}-h\eta^{2}+\frac{1}{2}p_{\eta}^{2}+\frac{1}{2}\frac{p_{\phi}^{2}}{\eta^{2}}.\label{eq:alpha2}\end{aligned}$$ These constants represent the conservation of a component of the generalised Runge-Lenz vector [@redmond_generalization_1964]. By inversion of $\alpha_{1}$ and $\alpha_{2}$ for $p_{\xi}$ and $p_{\eta}$, Hamilton’s equations finally yield $$\begin{aligned}
p_\xi=\frac{d\xi}{d\tau}&=\pm\frac{1}{\xi}\sqrt{\varepsilon\xi^{6}+2h\xi^{4}+2\alpha_{1}\xi^{2}-p_{\phi}^{2}},\label{eq:xi_diffeq}\\
p_\eta=\frac{d\eta}{d\tau}&=\pm\frac{1}{\eta}\sqrt{-\varepsilon\eta^{6}+2h\eta^{4}+2\alpha_{2}\eta^{2}-p_{\phi}^{2}},\label{eq:eta_diffeq}\\
\frac{d\phi}{d\tau}&=p_\phi\left(\frac{1}{\xi^2}+\frac{1}{\eta^2}\right).\label{eq:phi_diffeq}\end{aligned}$$ The solution of the Stark problem has thus been reduced to the integration by quadrature of eqs. –. Before proceeding, it is useful to outline the general features of the functions on the right-hand side of eqs. and .

Study of $p_{\xi}\left(\xi\right)$ and $p_{\eta}\left(\eta\right)$ {#subsec:poly_study}
------------------------------------------------------------------

Both $p_{\xi}\left(\xi\right)$ and $p_{\eta}\left(\eta\right)$ are functions of $\xi$ and $\eta$ symmetric with respect to both the horizontal and vertical axes. The zeroes of both functions are given by the roots of the bicubic polynomial radicands on the right-hand side of eqs. and . Hence, the number of real roots of $p_{\xi}\left(\xi\right)$ and $p_{\eta}\left(\eta\right)$ will depend on the initial conditions and on the physical parameters of the system (namely, the gravitational parameter and the value of the constant force field).

For any given set of initial conditions, it is clear that the polynomial radicand on 