 diagonal terms of this matrix are the eigenvalues of the Hamiltonian. The orthonormal eigenfunctions are expressed in terms of Jack symmetric polynomials [@jack69] which are very useful in determining the various physical properties of many particle systems with long-range interactions [@habook].

The search for an exact form of eigenfunction sometimes leads to partial diagonalization of the Hamiltonian [@tana05; @fin01]. Among the one dimensional systems with periodic boundary condition, several such quasi-solvable models exist. The eigenvalues and eigenfunctions for many of them have been obtained[@tur87; @ushbook]. The model with $sl(2)$ structure was first discovered by Turbiner and Ushveridze [@tur88]. It was also observed that the well known $N$ body Calogero-Sutherland models [@cal71; @suth71; @ruhl95] have similar Lie algebraic structure of $sl(N+1)$.

It may be noted that these models are in fact different generalizations of the classically integrable Inozemtsev model [@tana04; @ino83]. The common feature of these models with some underlying Lie algebraic structure is the existence of an invariant finite dimensional module of the associated Lie algebra. Post and Turbiner [@post95] studied a classification of linear differential operators of a single variable which have a finite dimensional invariant subspace spanned by monomials. One of the basic advantages of quasi-solvability is that, one can restrict the study to a finite dimensional submanifold of the full Hamiltonian. The finite dimensional matrix elements can be calculated by allowing the Hamiltonian to act on finite-dimensional subspaces of a Hilbert space on which it is originally defined. When the Hamiltonian operator preserves an infinite number of subsequences of such finite dimensional subspaces [@tana05] it becomes solvable. The exact solvability of a model is ensured when the closure property is imposed on the space on which the Hamiltonian is allowed to act.

In this article we study the integrability and solvability of a spinless non-rel