might have no real roots. Secondly, when the signs of the constants $\alpha_{1}$ and $\alpha_{2}$ are positive, $p_{\xi}$ and $p_{\eta}$ assume real values for $\xi=0$ and $\eta=0$, and the trajectories in the phase plane thus seemingly cross the vertical axes. Physically, the conditions $\xi=0$ and $\eta=0$ correspond (via eqs. and ) to polar transits (i.e., the test particle is passing through the negative or positive $z$ axis). But, according to the definition of parabolic coordinates, $\xi$ and $\eta$ are strictly non-negative quantities and thus the trajectories in the phase planes cannot enter the regions $\xi < 0$ and $\eta < 0$. In order to solve this apparent contradiction it can be shown how, in correspondence of a transit through $\xi = 0$ or $\eta = 0$, the corresponding momentum ($p_\xi$ or $p_\eta$) switches discontinuously its sign (and, concurrently, the azimuthal angle $\phi$ discontinuously changes by $\pm\pi$). In the phase plane, upon reaching the vertical axis from a positive $\xi$ or $\eta$, the trajectory will be discontinuously reflected with respect to the horizontal axis, and its evolution will proceed again towards positive $\xi$ or $\eta$. Figure \[fig:bidimensional\_case\] shows a selection of representative trajectories in the phase space for $p_{\xi}\left(\xi\right)$ and $p_{\eta}\left(\eta\right)$ 