According to the Bertrand theorem, the Kepler problem and the harmonic oscillator are the only central force dynamical systems that have closed orbits for all bounded motions. In this article, other dynamical systems having such a closed orbit property are found on T*(R3−{0}). Consider a natural dynamical system on T*(R4−{0}) whose Hamiltonian function is composed of kinetic and potential energies, and invariant under a SO(2) action. Then one can reduce the system to a Hamiltonian system on T*(R3−{0}) by the use of the Kustaanheimo–Stiefel transformation. If the original potential on R4−{0} is a central one, Bertrand’s method is applicable to the reduced system for determining the potential so that any bounded motions may be periodic. As a result, two types of potential functions will be found; one is linear in the radial variable and the other proportional to the inverse square root of that. The dynamical systems obtained are capable of physical interpretation. In particular, the dynamical system with the inverse square root potential may be called the twofold Kepler system, whose bounded trajectories have a self-intersection point.