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We investigate the matrix-model origin of the spherical sector of therational Calogero model and its constants of motion. We develop a diagrammatictechnique which allows us to find explicit expressions of the constants ofmotion and calculate their Poisson brackets. In this way we obtain allfunctionally independent constants of motion to any given order in the momenta.Our technique is related to the valence-bond basis for singlet states.

The spherical sector of the Calogero model as a reduced matrix model