Dynamics on the cone: Closed orbits and superintegrability

•Bertrand’s theorem is generalized to the case of the motion on a cone.•The superintegrability of the dynamics on a cone is discussed.•The WW-algebra of integrals of motion for Kepler and harmonic oscillator problems on a cone is derived.

The generalization of Bertrand’s theorem to the case of the motion of point particle on the surface of a cone is presented. The superintegrability of such models is discussed. The additional integrals of motion are analysed for the case of Kepler and harmonic oscillator potentials.