  Let $n\ge 2$ be an integer. To each irreducible representation $\sigma$ of
$\mathrm O(1)$, an $\mathrm {O}(1)$-Kepler problem in dimension $n$ is
constructed and analyzed. This system is super integrable and when $n=2$ it is
equivalent to a generalized MICZ-Kepler problem in dimension two. The dynamical
symmetry group of this system is $\widetilde {\mathrm{Sp}}_{2n}(\mathbb R)$
with the Hilbert space of bound states ${\mathscr H}(\sigma)$ being the unitary
highest weight representation of $\widetilde {\mathrm{Sp}}_{2n}(\mathbb R)$
with highest weight $$(\underbrace{-1/2, ..., -1/2}_{n-1}, -(1/2+|\sigma|)),$$
which occurs at the right-most nontrivial reduction point in the
Enright-Howe-Wallach classification diagram for the unitary highest weight
modules. (Here $|\sigma|=0$ or 1 depending on whether $\sigma$ is trivial or
not.) Furthermore, it is shown that the correspondence $\sigma\leftrightarrow
\mathscr H(\sigma)$ is the theta-correspondence for dual pair $(\mathrm{O}(1),
\mathrm{Sp}_{2n}(\mathbb R))\subseteq \mathrm{Sp}_{2n}(\mathbb R)$.
