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This fact was noticed by Moser (see ref. (4)).
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Olshanetsky, M.A., Perelomov, A.M. Explicit solution of the Calogero model in the classical case and geodesic flows on symmetric spaces of zero curvature. Lett. Nuovo Cimento 16, 333–339 (1976). https://doi.org/10.1007/BF02750226
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DOI: https://doi.org/10.1007/BF02750226