\right) = -\delta/\gamma$ will yield the same result in the right-hand side of eq. . The final result for $\phi$ is then[^6]: $$\begin{gathered}
\phi = \phi_0 + 2p_\phi\left\{\tau\left(\frac{1}{\gamma_\xi}+\frac{1}{\gamma_\eta}\right)+\frac{\delta_\xi-\beta_\xi\gamma_\xi}{\gamma_\xi^2\wp_\xi^\prime\left(u_\xi\right)}\right.\\
\cdot\left[\ln\frac{\sigma_\xi\left(\tau-\tau_\xi+u_\xi\right)}{\sigma_\xi\left(\tau-\tau_\xi-u_\xi\right)}
-\ln\frac{\sigma_\xi\left(-\tau_\xi+u_\xi\right)}{\sigma_\xi\left(-\tau_\xi-u_\xi\right)}-2\tau\zeta_\xi\left( u_\xi \right)\right]\\
+\frac{\delta_\eta-\beta_\eta\gamma_\eta}{\gamma_\eta^2\wp_\eta^\prime\left(u_\eta\right)}\\
\left.\cdot\left[\ln\frac{\sigma_\eta\left(\tau-\tau_\eta+u_\eta\right)}{\sigma_\eta\left(\tau-\tau_\eta-u_\eta\right)}
-\ln\frac{\sigma_\eta\left(-\tau_\eta+u_\eta\right)}{\sigma_\eta\left(-\tau_\eta-u_\eta\right)}
-2\tau\zeta_\eta\left( u_\eta \right)\right]
\vphantom{\frac{\tau}{\gamma_\xi}+\frac{\delta_\xi-\beta_\xi\gamma_\xi}{\gamma_\xi^2\wp_\xi^\prime\left(u_\xi\right)}}
\right\},\label{eq:phi_tau}\end{gathered}$$ where the following constants have been defined for notational convenience: $$\begin{aligned}
\beta_\xi & = -\frac{1}{24}f_{\xi}^{\prime\prime}\left(\frac{\xi_{r}^{2}}{2}\right),&\beta_\eta & = -\frac{1}{24}f_{\eta}^{\prime\prime}\left(\frac{\eta_{r}^{2}}{2}\right),\\
\gamma_\xi & = 2\xi_r^2, & \gamma_\eta & = 2\eta_r^2,\\
\delta_\xi & = f_\xi^\prime\left(\frac{\xi_r^2}{2}\right)+2\xi_r^2\beta_\xi,&\delta_\eta & = f_\eta^\prime\left(\frac{\eta_r^2}{2}\right)+2\eta_r^2\beta_\eta,\\
u_\xi&=\wp_\xi^{-1}\left(-\frac{\delta_\xi}{\gamma_\xi}\right),&u_\eta&=\wp_\eta^{-1}\left(-\frac{\delta_\eta}{\gamma_\eta}\right).\end{aligned}$$ Starting from eq. , we adopt the subscript notation $\sigma_\xi$ and $\zeta_\xi$ to indicate Weierstrass $\sigma$ and $\zeta$ functions defined in terms of the same invariants as $\wp_\xi$.

The time equation {#sec:time_equation}
=================

The final step in the solution of the Stark problem is to establish an explicit connection between real and fictitious time. To this end, we need to integrate eq. : $$dt=\left[\xi^{2}\left(\tau\right)+\eta^{2}\left(\tau\right)\right]d\tau.$$ In the general case, according to eqs. and , the exact solutions for $\xi^{2}\left(\tau\right)$ and $\eta^{2}\left(\tau\right)$ are of the form $$A+B\wp^{\prime}\left(\tau\right),\label{eq:wp_rational}$$ where A and B are rational functions of $\wp\left(\tau\right)$. Then, according to the theory of elliptic functions, the antiderivative of can be calculated in terms of $\wp$, $\wp^\prime$, $\wp^{-1}$ and the Weierstrass $\sigma$ and $\zeta$ functions. The integration method, due 