  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 $U(x) = 1/2 \omega^2 x^2$ or $U(x) = 1/8 \omega^2
x ^2 + c^2 x^{-2}.$
