  We study a class of Calogero-Sutherland type one dimensional N-body quantum
mechanical systems, with potentials given by $$ V( x_1, x_2, \cdots x_N) =
\sum_{i <j} {g \over {(x_i - x_j)^2}} - \frac{g^{\prime}}{\sum_{i<j}(x_i -
x_j)^2} + U(\sqrt{\sum_{i<j}(x_i - x_j)^2}),$$ where $U(\sqrt{\sum_{i<j}(x_i -
x_j)^2})$'s are of specific form. It is shown that, only for a few choices of
$U$, the eigenvalue problems can be solved {\it exactly}, for arbitrary
$g^{\prime}$. The eigen spectra of these Hamiltonians, when $g^{\prime} \ne 0$,
are non-degenerate and the scattering phase shifts are found to be energy
dependent. It is further pointed out that, the eigenvalue problems are amenable
to solution for wider choices of $U$, if $g^{\prime}$ is conveniently fixed.
These conditionally exactly solvable problems also do not exhibit energy
degeneracy and the scattering phase shifts can be computed {\it only} for a
specific partial wave.
