A note on quantum Bohlin transformation

It is shown, that the reduction of the circular quantum oscillator by the $Z_2$-group action results to the two systems: a two-dimensional hydrogen atom, and a ``charge - charged magnetic vortex" one, with the spin $\frac 12$. Analogously, the $Z_N$-reduction of the two-dimensional system with the central potential $r^{2(N-1)}$ results into $N$ bound ``charge - magnetic vertex" systems with the interaction potential $r^{2(1/N-1)}$ and spins $\sigma=\frac kN$, $k =0,1,..., (N-1)$.