= First integral of a cubic-quadratic reversible flow
{title2=$I=|F_+|^{-a_-}|F_-|^{a_+}$}
For $\dot x=y^3-y$, $\dot y=x-y^2$, set $a_\pm=(1\pm\sqrt3)/2$ and $F_\pm=x-1-a_\pm(y^2-1)$. Direct differentiation gives $\dot F_\pm=-2a_\pm yF_\pm$, proving the displayed first integral. The invariant parabolas meet at saddles $(1,\pm1)$ 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.
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