  The $2N$-dimensional quantum problem of $N$ particles (e.g. electrons) with
interaction $\beta/r^2$ in a two-dimensional parabolic potential $\omega_0$
(e.g. quantum dot) and magnetic field $B$, reduces exactly to solving a
$(2N-4)$-dimensional problem which is independent of $B$ and $\omega_0$. An
exact, infinite set of relative mode excitations are obtained for any $N$. The
$N=3$ problem reduces to that of a ficticious particle in a two-dimensional,
non-linear potential of strength $\beta$, subject to a ficticious magnetic
field $B_{\rm fic}\propto J$, the relative angular momentum.
