  We analyze the periodic motion in the conformal mechanics describing the
particles moving near the horizon of extreme Reissner-Nordstr\"om and
axion-dilaton (Cl\'ement-Gal'tsov) black holes. For this purpose we extract the
(two-dimensional) compact ("angular") parts of these systems and construct
their action-angle variables. In the first case we get the well-known spherical
Landau problem, which possesses hidden $so(3)$ symmetry, while in the latter
case the system does not have hidden constant of motion. In both cases we
indicate the existence of "critical points", separating the regions of periodic
motions with qualitatively different properties.
