Some new trends in superintegrable systems

The Hamiltonian integrability in classical mechanics and the explicit metrics for a class of two-dimensional cubically superintegrable systems are reviewed. Firstly, the integrals that are quadratic in moments corresponding to the natural Lagrangian systems are discussed with a special view focus to the two-dimensional case. The classical free Lagrangian admitting a constant of motion, in one and two dimensional space, was generalized by using the fractional Caputo derivative. The fractional Killing vectors and Killing-Yano tensors are presented in connection with the hidden symmetries of curved spaces. The Dunkl-Coulomb system in the plane was considered. The model that was defined in terms of the Dunkl Laplacian, involves reflection operators, with a r^(-1) potential. The system is shown to be maximally superintegrable and exactly solvable.