  We develop an algebraic approach for finding the eigenfunctions of a large
class of few and many-body Hamiltonians, in one and higher dimensions, having
linear spectra. The method presented enables one to exactly map these
interacting Hamiltonians to decoupled oscillators, thereby, giving a precise
correspondence between the oscillator eigenspace and the wavefunctions of these
quantum systems. The symmetries behind the degeneracy structure and the
commuting constants of motion responsible for the quantum integrability of some
of these models are made transparent. Apart from analysing a number of
well-known dynamical systems like planar oscillators with commensurate
frequencies, both with or without singular inverse-square terms and generalized
Calogero-Sutherland type models, we also point out a host of other examples
where the present approach can be profitably employed. We further study
Hamiltonians having Laughlin wavefunction as the ground-state and establish
their equivalence to free oscillators. This reveals the underlying linear
$W_{1+\infty}$ symmetry algebra unambiguously and establishes the Laughlin
wavefunction as the highest weight vector of this algebra.
