 Solution

The equation has the one-dimensional effective potential per unit massThus the conserved specific energy isThe potential has its minimum $-2mk/a$ at $z=0$ and tends to $0$ from below as $|z|\to \infty$. Therefore $-2mk/a<\mathcal{E}<0$ gives bounded oscillation between two turning points; $\mathcal{E}=0$ gives marginal escape with asymptotic speed zero; and $\mathcal{E}>0$ gives escape with nonzero asymptotic speed. The minimum itself is the equilibrium $z=0$.