the previous section, the polynomials $f_\xi$ and $f_\eta$ will always have at least one positive real root, with the exception of the bidimensional case for $f_\xi$ with $\alpha_1 > 0$ (displayed in Figure \[fig:bidimensional\_case\] (b)). The roots $\xi_r$ and $\eta_r$ of the cubic polynomials $f_\xi$ and $f_\eta$ can be computed exactly using the general formulæ for the roots of a cubic function. Secondly, in order to determine $\tau_\xi$ (and, analogously, $\tau_\eta$) we can use eq. to write: $$\tau_\xi=\pm\int_{\frac{1}{2}\xi_{0}^{2}}^{\frac{1}{2}\xi_r^{2}}\frac{ds}{\sqrt{8\varepsilon s^{3}+8hs^{2}+4\alpha_{1}s-p_{\phi}^{2}}}.
\label{eq:depressed_transform}$$ Following then @byrd_handbook_1971 [eqs. (A7)–(A13)], we introduce the Tschirnhaus transformation [@cayley_tschirnhausens_1861] $$s = \sqrt[3]{\frac{1}{2\varepsilon}}s_1 - \frac{1}{3}\frac{h}{\varepsilon}$$ in order to reduce the polynomial $f_\xi$ to a depressed cubic: $$\tau_\xi=\pm\sqrt[3]{\frac{1}{2\varepsilon}}\int_{\sqrt[3]{2\varepsilon}\left(\frac{1}{2}\xi_0^2+\frac{1}{3}\frac{h}{\varepsilon}\right)}^
{\sqrt[3]{2\varepsilon}\left(\frac{1}{2}\xi_r^2+\frac{1}{3}\frac{h}{\varepsilon}\right)}
\frac{ds_1}{
\sqrt{4s_{1}^{3}-h_2s_{1}-h_3}},$$ where $$\begin{aligned}
h_2 & = \sqrt[3]{\frac{1}{2\varepsilon}}\left(\frac{8}{3}\frac{h^{2}}{\varepsilon}-4\alpha_{1}\right), \\
h_3 & = \frac{4}{3}\frac{\alpha_{1}h}{\varepsilon}-\frac{16}{27}\frac{h^{3}}{\varepsilon^{2}}+p_{\phi}^{2}.\end{aligned}$$ The integral can now be split into two separate Weierstrass normal elliptic integrals of the first kind, $$\begin{gathered}
\tau_\xi=\pm\sqrt[3]{\frac{1}{2\varepsilon}}\left[
\int_{\sqrt[3]{2\varepsilon}\left(\frac{1}{2}\xi_0^2+\frac{1}{3}\frac{h}{\varepsilon}\right)}^{\infty}
\frac{ds_1}{\sqrt{4s_{1}^{3}-h_2s_{1}-h_3}}
\right.\\
\left.
-\int_{\sqrt[3]{2\varepsilon}\left(\frac{1}{2}\xi_r^2+\frac{1}{3}\frac{h}{\varepsilon}\right)}^{\infty}
\frac{ds_1}{\sqrt{4s_{1}^{3}-h_2s_{1}-h_3}}
\right],\end{gathered}$$ and solved in terms of the inverse Weierstrass elliptic function $\wp^{-1}$ as $$\begin{gathered}
\tau_\xi=\pm\sqrt[3]{\frac{1}{2\varepsilon}}\left\{
\wp^{-1}\left[\sqrt[3]{2\varepsilon}\left(\frac{1}{2}\xi_0^2+\frac{1}{3}\frac{h}{\varepsilon}\right);h_2,h_3\right]\right.\\
\left.-\wp^{-1}\left[\sqrt[3]{2\varepsilon}\left(\frac{1}{2}\xi_r^2+\frac{1}{3}\frac{h}{\varepsilon}\right);h_2,h_3\right]
\right\}.\label{eq:tau_xi}\end{gathered}$$ The corresponding formula for $\tau_\eta$ can be obtained by switching $\varepsilon$ to $-\varepsilon$ and $\alpha_1$ to $\alpha_2$. It must be noted though that in this formula there are ambiguities regarding the computation of the inverse 