Fractional extensions of the classical isotropic oscillator and the Kepler problem

The class of fractional Hamiltonian systems that generalize the classical problem of the two-dimensional (2D) isotropic harmonic oscillator and the Kepler problem is considered. It is shown that, in the 4D space of structural parameters, the 2D isotropic harmonic oscillator can be extended along a line in such a way that the orbits remain closed and oscillations remain isochronous. Likewise, the Kepler problem can be extended along a line in such a way that the orbits remain closed for all finite motions and the third Kepler law remains valid. These curves lie on the 2D surfaces where any dynamical system is characterized by the same rotation number for all finite motions.