We make a new multivariate generalization of the type A monomial space of a single variable. It is different from the previously introduced type A space of several variables which is an sl(M+1) module, and we thus call it type A'. We construct the most general quasi-solvable operator of (at most) second-order which preserves the type A' space. Investigating directly the condition under which the type A' operators can be transformed to Schroedinger operators, we obtain the complete list of the type A' quasi-solvable quantum many-body systems. In particular, we find new quasi-solvable models of deformed Calogero-Sutherland type which are different from the Inozemtsev systems. We also examine a new multivariate generalization of the type C monomial space based on the type A' scheme.
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