  A similarity transformation is constructed through which a system of
particles interacting with inverse-square two-body and harmonic potentials in
one dimension, can be mapped identically, to a set of free harmonic
oscillators. This equivalence provides a straightforward method to find the
complete set of eigenfunctions, the exact constants of motion and a linear
$W_{1+\infty}$ algebra associated with this model. It is also demonstrated that
a large class of models with long-range interactions, both in one and higher
dimensions can be made equivalent to decoupled oscillators.
