= First integral of the quadratic fold-Hopf amplitude flow
{title2=$F=-u(\lambda_2+u^2/3+v^2)$}
The amplitude equations $u'=-2uv$, $v'=\lambda_2+v^2+u^2$ satisfy $u'=F_v$, $v'=-F_u$, hence preserve the displayed <first integral>. For $\lambda_2=-a^2<0$ and $u\ge0$, the <centre equilibrium> is $(a,0)$ with $F=2a^3/3$, while the two boundary <saddle equilibria> are $(0,\pm a)$ with $F=0$. The level $F=0$ contains the boundary connection and the arc $u^2=3(a^2-v^2)$ joining the two <saddle equilibria>. Levels $0<F<2a^3/3$ form closed <periodic orbits>. Adding an independent rotating phase turns these closed amplitude trajectories into invariant <tori>.
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