  The monopole systems with hidden symmetry of the two-dimensional Coulomb
problem are considered.
  One of them, the "charge-charged magnetic vortex" with a half-spin, is
constructed by reducing the quantum circular oscillator with respect to the
action of the parity operator.
  The other two systems are constructed by reduction from the two-dimensional
complex space. The first system is a particle on the sphere in the presence of
the exterior constant magnetic field (generated by Dirac's monopole located in
its center). This system is dual to the massless (3+1)-dimensional particle
with fixed energy. The second system represents the particle on the
pseudosphere in the presence of exterior magnetic field and is dual to the
massive relativistic anyon.
