  We propose an exactly-solvable model of the quantum oscillator on the class
of K\"ahler spaces (with conic singularities), connected with two-dimensional
complex projective spaces. Its energy spectrum is nondegenerate in the orbital
quantum number, when the space has non-constant curvature. We reduce the model
to a three-dimensional system interacting with the Dirac monopole. Owing to
noncommutativity of the reduction and quantization procedures, the Hamiltonian
of the reduced system gets non-trivial quantum corrections. We transform the
reduced system into a MIC-Kepler-like one and find that quantum corrections
arise only in its energy and coupling constant. We present the exact spectrum
of the generalized MIC-Kepler system. The one-(complex) dimensional analog of
the suggested model is formulated on the Riemann surface over the complex
projective plane and could be interpreted as a system with fractional spin.
