  A nonrelativistic particle on a circle and subject to a cos^{-2}(k phi)
potential is related to the two-dimensional (dihedral) Coxeter system I_2(k),
for k in N. For such `dihedral systems' we construct the action-angle variables
and establish a local equivalence with a free particle on the circle. We
perform the quantization of these systems in the action-angle variables and
discuss the supersymmetric extension of this procedure. By allowing radial
motion one obtains related two-dimensional systems, including A_2, BC_2 and G_2
three-particle rational Calogero models on R, which we also analyze.
