Periodic orbits for an infinite family of classical superintegrable systems

We show that all bounded trajectories in the two dimensional classical system with the potential $V(r,ϕ)=ω^2 r^2+ \frac{\al k^2}{r^2 \cos^2 {k ϕ}}+ \frac{βk^2}{r^2 \sin^2 {k ϕ}}$ are closed for all integer and rational values of $k$. The period is $T=\fracπ{2ω}$ and does not depend on $k$. This agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable.

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