the $\omega_i$ are the fundamental half-periods of $\wp$ and the third one is the sum of the fundamental half-periods). The solution of eq. is thus: $$\begin{gathered}
t = \left(\xi_r^2+\eta_r^2\right)\tau+\frac{1}{2}\frac{f_{\xi}^{\prime}\left(\frac{\xi_{r}^{2}}{2}\right)}{g_{2,\xi}/4-3e_{i,\xi}^2}\cdot
\\\left[\tau e_{i,\xi} + \zeta_\xi\left(\tau-\tau_\xi-\omega_{i,\xi}\right)-\zeta_\xi\left(-\tau_\xi-\omega_{i,\xi}\right)\right]+\\
\frac{1}{2}\frac{f_{\eta}^{\prime}\left(\frac{\eta_{r}^{2}}{2}\right)}{g_{2,\eta}/4-3e_{i,\eta}^2}\cdot\\
\left[\tau e_{i,\eta} + \zeta_\eta\left(\tau-\tau_\eta-\omega_{i,\eta}\right)-\zeta_\eta\left(-\tau_\eta-\omega_{i,\eta}\right)\right].
\label{eq:time_equation}\end{gathered}$$ This equation can be considered as the equivalent of Kepler’s equation for the Stark problem. Similarly to the two-body problem, it is constituted of a linear part modulated by two quasi-periodic transcendental parts (with the Weierstrass $\zeta$ function replacing the sine function appearing in Kepler’s equation). In this sense, the fictitious time $\tau$ can be regarded as a kind of eccentric anomaly for the Stark problem. According to eq. , the time equation is a monotonic function and its inversion can thus be achieved numerically using standard techniques (Newton-Raphson, bisection, etc.).

Analysis of the results
=======================

After having determined the full formal solution of the Stark problem in the previous sections, we now turn our attention to the interpretation of the results.

Before proceeding, we first need to point out how our solution to the Stark problem, as developed in the previous sections, is directly applicable to the three-dimensional case, but not in general to all bidimensional cases. As explained in 