Modification of Bertrand’s Theorem and Extended Runge-Lenz Vector

It is shown that for a particle with suitable angular momenta in the screened Coulomb potential or isotropic harmonic potential, there still exist closed orbits rather than ellipse, characterized by the conserved aphelion and perihelion vectors, i.e. extended Runge-Lenz vector, which implies a higher dynamical symmetry than the geometrical symmetry O₃. The closeness of a planar orbit implies the radial and angular motional frequencies are commensurable.