= Solution
For $k=0$, the <first integral>
$$
E(x,y)=\frac{y^2}{2}+\frac{x^2}{2}-\frac{x^3}{3}
$$
has $\dot E=0$. The <equilibrium points> are $(0,0)$ and $(1,0)$. The <linearization> has <eigenvalues> $\pm i$ at the origin and $\pm1$ at $(1,0)$. The origin is a <center equilibrium>: the sufficiently small positive <level sets> of $E$ are closed curves, since its <Hessian matrix> there is <positive-definite>. The other equilibrium is a <saddle equilibrium>.
At the saddle, $E=1/6$, and the <separatrix> satisfies
$$
y^2=\frac{(x-1)^2(2x+1)}3.
$$
Its two branches with $-1/2\leq x\leq1$ form a <homoclinic orbit> through the saddle; its remaining branches extend to $x>1$. Inside the loop, the closed trajectories run clockwise: above the horizontal axis $\dot x=y>0$. The saddle directions have slopes $+1$ and $-1$. Outside the loop there are unbounded trajectories. The left panel shows the closed orbits and all saddle branches.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-1-phase-planes.png]
{title=Conservative phase portrait and the attracting basin after adding weak damping}
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