Abstract
We define the ``maximally integrable'' isotropic oscillator on ${\mathrm{CP}}^{N}$ and discuss its various properties, in particular, the behavior of the system with respect to a constant magnetic field. We show that the properties of the oscillator on ${\mathrm{CP}}^{N}$ qualitatively differ in the $N>1$ and $N=1$ cases. In the former case we construct the ``axially symmetric'' system which is locally equivalent to the oscillator. We perform the Kustaanheimo-Stiefel transformation of the oscillator on ${\mathrm{CP}}^{2}$ and construct some generalized MIC-Kepler problem. We also define a $\mathcal{N}=2$ superextension of the oscillator on ${\mathrm{CP}}^{N}$ and show that for $N>1$ the inclusion of a constant magnetic field preserves the supersymmetry of the system.
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