 First integral of a cubic-quadratic reversible flow

For $\dot{x}=y^3-y$, $\dot{y}=x-y^2$, set $a_{\pm }=(1\pm \sqrt{3})/2$ and $F_{\pm }=x-1-a_{\pm }(y^2-1)$. Direct differentiation gives ${\dot{F}}_{\pm }=-2a_{\pm }y{F}_{\pm }$, proving the displayed first integral. The invariant parabolas meet at saddles $(1,\pm 1)$ and bound a heteroclinic cycle. Inside that lens, the logarithm of $I$ has a strict nondegenerate maximum at the origin, proving a nonlinear center equilibrium, rather than inferring one solely from imaginary Jacobian eigenvalues.