a portion of trajectory in which $p_\xi$ remains positive ($+$) or negative ($-$).

Changing now integration variable in the right-hand side of eq. to $$s=\frac{1}{2}u^{2},$$ we can rewrite the equation as $$\begin{aligned}
\tau&=\pm\int_{\frac{1}{2}\xi_{0}^{2}}^{\frac{1}{2}\xi^{2}}\frac{ds}{\sqrt{8\varepsilon s^{3}+8hs^{2}+4\alpha_{1}s-p_{\phi}^{2}}} \label{eq:tau_t}\\
&=\pm\int_{\frac{1}{2}\xi_{0}^{2}}^{\frac{1}{2}\xi^{2}}\frac{ds}{\sqrt{f_{\xi}\left(s\right)}},\end{aligned}$$ where $f_{\xi}\left(s\right)$ is a cubic polynomial in $s$. The integral in this expression is an elliptic integral, which can be computed and inverted to yield $\xi^{2}$ as function of $\tau$ using a formula by Weierstrass [see @whittaker_course_1927 §20.6]. After electing $$\begin{aligned}
f_{\xi}\left(s\right) & =a_{4}+4a_{3}s+6a_{2}s^{2}+4a_{1}s^{3},\end{aligned}$$ and defining $$\begin{aligned}
g_{2} & =-4a_{1}a_{3}+3a_{2}^{2},\label{eq:g2_def}\\
g_{3} & =2a_{1}a_{2}a_{3}-a_{2}^{3}-a_{1}^{2}a_{4},\label{eq:g3_def}\\
\wp_{\xi}\left(\tau\right) & \equiv\wp\left(\tau;g_{2},g_{3}\right),\end{aligned}$$ where $\wp\left(\tau;g_{2},g_{3}\right)$ is a Weierstrass elliptic function defined in terms of the invariants $g_{2}$ and $g_{3}$ (see @whittaker_course_1927, Chapter XX, and @abramowitz_handbook_1964, Chapter 18), the evolution of $\xi^{2}$ in fictitious time is given by $$\begin{aligned}
\xi^{2} & =\xi_{0}^{2}+\frac{1}{\left[\wp_{\xi}\left(\tau\right)-\frac{1}{24}f_{\xi}^{\prime\prime}\left(\frac{\xi_{0}^{2}}{2}\right)\right]^{2}}\nonumber \\
 & \quad\cdot\left\{ \frac{1}{2}f_{\xi}^{\prime}\left(\frac{\xi_{0}^{2}}{2}\right)\left[\wp_{\xi}\left(\tau\right)-\frac{1}{24}f_{\xi}^{\prime\prime}\left(\frac{\xi_{0}^{2}}{2}\right)\right]\vphantom{\sqrt{f_{\xi}\left(\frac{\xi_{0}^{2}}{2}\right)}}\right.\nonumber \\
 & \quad\left.+\frac{1}{24}f_{\xi}\left(\frac{\xi_{0}^{2}}{2}\right)f_{\xi}^{\prime\prime\prime}\left(\frac{\xi_{0}^{2}}{2}\right)\pm\sqrt{f_{\xi}\left(\frac{\xi_{0}^{2}}{2}\right)}\wp_{\xi}^{\prime}\left(\tau\right)\right\}.\label{eq:xi_tau}\end{aligned}$$ Here the $\pm$ sign represents the sign ambiguity discussed earlier, and the derivatives of $f_\xi$ are calculated with respect to the polynomial variable, while the derivative $\wp_\xi^\prime$ is calculated with respect to $\tau$. $\wp_\xi^\prime$ is related to $\wp$ via the relation $$\left[\wp_\xi^{\prime}\left( z \right)\right] ^2 = 4 \wp_\xi^3\left( z \right) - g_2 \wp_\xi\left( z \right) - g_3 \label{eq:Pprime_rel}$$ [@abramowitz_handbook_1964 eq. 18.1.6]. If $\xi_{0}^{2}/2$ 