= Solution
The line $x=0$ is invariant for every $a$; on it, $\dot y=1-y^2$, so the segment from $P_-$ to $P_+$ is a heteroclinic orbit. For $a=0$, the zero-energy set is
$$
H=0
\quad\Longleftrightarrow\quad
x=0\quad\text{or}\quad y^2+\frac{x^2}{3}=1.
$$
These separatrices divide the two families of closed orbits around the centers from the exterior trajectories. For $a=1/2$, both axial equilibria are saddles and $Q_\pm=(\pm\sqrt3/2,1/2)$ are stable foci. For $a=3/2$, the off-axis equilibria have disappeared, $P_+$ is a stable node, and $P_-$ remains a saddle.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-1-dynamical-system-phase-portraits.png]
{title=<Phase portraits> of the planar <dynamical system>}
The arrows, fixed-point types, invariant axis, and the exact $a=0$ separatrix are shown in the three <phase portraits>.
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