A family of complete integrable three-dimensional N-body quantum systems is introduced and completely solved by the dynamical algebras O(3N+1,2) and their representations. In a particular realisation of these algebras the particles interact by N-body 'Coulomb-type' potentials. A complete set of commuting integrals of motions, their spectra, the energy levels for both discrete and continuous spectra and their degeneracy have been explicitly determined. Relativistic generalisations and applications are briefly discussed.
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Barut et al. (1981) studied this question.
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