Abstract
The classification of maximal abelian subgroups of the noncompact Lie group SU(p, q) is reviewed and then applied to construct maximally superintegrable Hamiltonian systems on real hyperboloids. These Hamiltonian systems are obtained by projecting free motion on the homogeneous space SU (p, q) /U (p -1, q) onto the space O (p, q)/O (p -1, q). The projection is realized by introducing ignorable variables, corresponding to different maximal abelian subalgebras of su (p,q). Classical and quantum mechanical systems are treated on the same footing.
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