  For each integer $n\ge 1$, we demonstrate that a $(2n+1)$-dimensional
generalized MICZ-Kepler problem has an $\mr{Spin}(2, 2n+2)$ dynamical symmetry
which extends the manifest $\mr{Spin}(2n+1)$ symmetry. The Hilbert space of
bound states is shown to form a unitary highest weight $\mr{Spin}(2,
2n+2)$-module which occurs at the first reduction point in the
Enright-Howe-Wallach classification diagram for the unitary highest weight
modules. As a byproduct, we get a simple geometric realization for such a
unitary highest weight $\mr{Spin}(2, 2n+2)$-module.
