  We propose a systematic method to construct quasi-solvable quantum many-body
systems having permutation symmetry. By the introduction of elementary
symmetric polynomials and suitable choice of a solvable sector, the algebraic
structure of sl(M+1) naturally emerges. The procedure to solve the
canonical-form condition for the two-body problem is presented in detail. It is
shown that the resulting two-body quasi-solvable model can be uniquely
generalized to the M-body system for arbitrary M under the consideration of the
GL(2,K) symmetry. An intimate relation between quantum solvability and
supersymmetry is found. With the aid of the GL(2,K) symmetry, we classify the
obtained quasi-solvable quantum many-body systems. It turns out that there are
essentially five inequivalent models of Inozemtsev type. Furthermore, we
discuss the possibility of including M-body (M>=3) interaction terms without
destroying the quasi-solvability.
