  Let $n\ge 2$ be a positive integer. To each irreducible representation
$\sigma$ of $\mathrm{Sp}(1)$, an $\mathrm{Sp}(1)$-Kepler problem in dimension
$(4n-3)$ is constructed and analyzed. This system is super integrable and when
$n=2$ it is equivalent to a generalized MICZ-Kepler problem in dimension five.
The dynamical symmetry group of this system is $\widetilde {\mathrm O}^*(4n)$
with the Hilbert space of bound states ${\mathscr H}(\sigma)$ being the unitary
highest weight representation of $\widetilde {\mathrm {O}^*}(4n)$ with highest
weight $$(\underbrace{-1, ..., -1}_{2n-1}, -(1+\bar\sigma)),$$ which occurs at
the right-most nontrivial reduction point in the Enright-Howe-Wallach
classification diagram for the unitary highest weight modules. Here
$\bar\sigma$ is the highest weight of $\sigma$. Furthermore, it is shown that
the correspondence $\sigma\leftrightarrow \mathscr H(\sigma)$ is the
theta-correspondence for dual pair $(\mathrm{Sp}(1),
\mathrm{O}^*(4n))\subseteq\mathrm{Sp}_{8n}(\mathbb R)$.
