Abstract
We define the "maximally integrable" isotropic oscillator on CP(N) and discuss its various properties, in particular, the behaviour of the system with respect to a constant magnetic field. We show that the properties of the oscillator on 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 CP(2) and construct some generalized MIC-Kepler problem. We also define a N=2 superextension of the oscillator on 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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