A remark on rational isochronous potentials

We consider the rational potentials of the one-dimensional mechanical systems, which have a family of periodic solutions with the same period (isochronous potentials). We prove that up to a shift and adding a constant all such potentials have the form $U(x) = 1/2 ω^2 x^2$ or $U(x) = 1/8 ω^2 x ^2 + c^2 x^{-2}.$

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