  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.
