= Closed orbits with an inverse-cube perturbation
{title2=$\nu=\sqrt{1-2\mu a/L^2}\in\mathbb Q_{>0}$}
In the <gravitational potential> $-\mu(1/r+a/r^2)$, a regular noncircular bound <orbit> requires $L^2>2\mu a$ and has $u=\mu(L^2-2\mu a)^{-1}[1+e\cos\nu(\psi-\psi_0)]$ with $0<e<1$. Each radial period advances the angle by $2\pi/\nu$. The full position and <velocity> return after finitely many radial periods precisely when $\nu$ is rational. Irrational $\nu$ gives a dense rosette on the two-frequency <invariant torus>. Rationality permits a fifth <orbit> label fixing the apsidal phase on that resonant <angular momentum> family, but does not give global superintegrability for all nearby angular momenta. When $a=0$, every regular bound <Kepler orbit> is closed and the <Laplace-Runge-Lenz vector> supplies the globally defined additional integral.
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