We split the generic conformal mechanical system into a "radial" and an "angular" part, where the latter is defined as the Hamiltonian system on the orbit of the conformal group, with the Casimir function in the role of the Hamiltonian. We reduce the analysis of the constants of motion of the full system to the study of certain differential equations on this orbit. For integrable mechanical systems, the conformal invariance renders them superintegrable, yielding an additional series of conserved quantities originally found by Wojciechowski in the rational Calogero model. Finally, we show that, starting from any N=4 supersymmetric "angular" Hamiltonian system one may construct a new system with full N=4 superconformal D(1,2;\alpha) symmetry.Comment: 9 pages revte

Oscillator and Coulomb systems on N -dimensional spaces of constant curvature can be generalized by replacing their angular degrees of freedom with a compact integrable (N −1)-dimensional system. We present the actionangle formulation of such models in terms of the radial degree of freedom and the action-angle variables of the angular subsystem. As an example, we construct the spherical and pseudospherical generalization of the two-dimensional superintegrable models introduced by Tremblay, Turbiner and Winternitz and by Post and Winternitz. We demonstrate the superintegrability of these systems and give their hidden constant of motion.

We exclude the center of mass of the N -particle rational Calogero model and consider the angular part of the resulting Hamiltonian. We show that it describes the motion of the particle on (N − 2)dimensional sphere interacting with N (N − 1)/2 force centers with Higgs oscillator potential. In the case of four-particle system these force centers define the vertexes of an Archimedean solid called cuboctahedron. * Electronic address: hakob@yerphi.am † Electronic address: arnerses@yerphi.am