is chosen as a root $\xi_{r}^{2}/2$ of $f_{\xi}$, then $f_\xi\left( \xi_r^2/2 \right) = 0$ and eq. simplifies to $$\xi^{2}=\xi_{r}^{2}+\frac{1}{2}\frac{f_{\xi}^{\prime}\left(\frac{\xi_{r}^{2}}{2}\right)}{\wp_{\xi}\left(\tau-\tau_{\xi}\right)-\frac{1}{24}f_{\xi}^{\prime\prime}\left(\frac{\xi_{r}^{2}}{2}\right)},
\label{eq:xi_simpl}$$ where $\tau_{\xi}$ is the fictitious time for which $\xi$ assumes the value $\xi_{r}$. The analogous expressions for $\eta$ are $$\begin{aligned}
\eta^{2} & =\eta_{0}^{2}+\frac{1}{\left[\wp_{\eta}\left(\tau\right)-\frac{1}{24}f_{\eta}^{\prime\prime}\left(\frac{\eta_{0}^{2}}{2}\right)\right]^{2}}\nonumber \\
 & \quad\cdot\left\{ \frac{1}{2}f_{\eta}^{\prime}\left(\frac{\eta_{0}^{2}}{2}\right)\left[\wp_{\eta}\left(\tau\right)-\frac{1}{24}f_{\eta}^{\prime\prime}\left(\frac{\eta_{0}^{2}}{2}\right)\right]\vphantom{\sqrt{f_{\eta}\left(\frac{\eta_{0}^{2}}{2}\right)}}\right.\nonumber \\
 & \quad\left.+\frac{1}{24}f_{\eta}\left(\frac{\eta_{0}^{2}}{2}\right)f_{\eta}^{\prime\prime\prime}\left(\frac{\eta_{0}^{2}}{2}\right)\pm\sqrt{f_{\eta}\left(\frac{\eta_{0}^{2}}{2}\right)}\wp_{\eta}^{\prime}\left(\tau\right)\right\} \label{eq:eta_tau}\end{aligned}$$ and $$\eta^{2}=\eta_{r}^{2}+\frac{1}{2}\frac{f_{\eta}^{\prime}\left(\frac{\eta_{r}^{2}}{2}\right)}{\wp_{\eta}\left(\tau-\tau_{\eta}\right)-\frac{1}{24}f_{\eta}^{\prime\prime}\left(\frac{\eta_{r}^{2}}{2}\right)}.
\label{eq:eta_simpl}$$ The formulæ and represent a general and complete closed-form solution for the squares $\xi^2$ and $\eta^2$ of the parabolic coordinates $\xi$ and $\eta$. Since $\xi$ and $\eta$ are non-negative by definition, in order to recover the solution for $\xi$ and $\eta$ it will be enough to take the principal square root of $\xi^2$ and $\eta^2$. The cartesian positions and velocities can be reconstructed using eqs. -, where the derivatives of the parabolic coordinates with respect to the real time can be computed by inverting eqs. - (and by keeping in mind that $p_\xi$ and $p_\eta$ can be calculated by differentiating eqs. and with respect to $\tau$ – see eqs. and ).

For simplicity’s sake, notational convenience and further analysis, however, it is desirable to be able to use the simplified formulæ and whenever possible. To this end, we first note how, from the considerations presented in 