On the Zoll deformations of the Kepler problem

A celebrated result of Bertrand states that the only central force potentials on the plane with the property that all bounded orbits are periodic are the Kepler potential and the potential of the harmonic oscillator. In this paper, we complement Bertrand's theorem showing the existence of an infinite-dimensional space of central force potentials on the plane which are Zoll at a given energy level, meaning that all non-collision orbits with given energy are closed and of the same length. We also determine all natural systems on the (not necessarily flat) plane which are invariant under rotations and Zoll at a given energy and prove several existence and rigidity results for systems which are Zoll at multiple energies.