 Solution

Dividing the two velocities givesHere $\dot{z}=a{x}^2/m\ge 0$: the orbit always advances toward increasing $z$, except for instantaneous zeros at each midplane crossing $x=0$. The transverse motion oscillates between $-X$ and $X$. At these outer turning points $\dot{x}=0$ and the trajectory is tangent to the $z$ direction; at the midplane $\dot{x}\ne 0$ while $\dot{z}=0$, so the tangent is along $x$. The sign of the initial $\dot{x}$ only chooses the phase or reflected trace.