the right-hand side of eq. will tend to $+\infty$ for $\xi\to\pm\infty$, since $\varepsilon>0$ by definition. Thus, $p_{\xi}\left(\xi\right)$ will always tend to $\pm\infty$ in the limit $\xi\to\pm\infty$. Conversely, for $\eta\to\pm\infty$, the radicand in $p_{\eta}\left(\eta\right)$ will eventually start assuming negative values, thus implying the existence of a real root. For both $p_{\xi}\left(\xi\right)$ and $p_{\eta}\left(\eta\right)$, moving along the horizontal axis towards the origin from the initial conditions means encountering another root, as for $\xi=\eta=0$ both functions result in the computation of the square root of the negative value $-p_{\phi}^{2}$. This also implies that, in the three-dimensional problem, the trajectories in the phase planes $\left(\xi,p_\xi \right)$ and $\left(\eta,p_\eta\right)$ will not cross the vertical axes, and $p_{\xi}\left(\xi\right)$ and $p_{\eta}\left(\eta\right)$ always have at least two real roots. Figure \[fig:tridimensional\_case\] shows a selection of representative trajectories in the phase space for $p_{\xi}\left(\xi\right)$ and $p_{\eta}\left(\eta\right)$ in the three-dimensional case.

The bidimensional case requires a separate analysis. When $p_{\phi}$ is null, the bicubic polynomials collapse to biquadratic polynomials (via the inclusion of the external factors $1/\xi$ and $1/\eta$). As in the three-dimensional case, the evolution of $p_{\xi}$ can be either bound or unbound, while the evolution of $p_{\eta}$ is always bound. The first difference is that, when $\alpha_{1}>0$, $p_{\xi}$ 