  For the Jordan algebra of hermitian matrices of order $n\ge 2$, we let $X$ be
its submanifold consisting of rank-one semi-positive definite elements. The
composition of the cotangent bundle map $\pi_X$: $T^*X\to X$ with the canonical
map $X\to \mathbb{C}P^{n-1}$ (i.e., the map that sends a hermitian matrix to
its column space), pulls back the K\"{a}hler form of the Fubini-Study metric on
$\mathbb{C}P^{n-1}$ to a real closed differential two-form $\omega_K$ on
$T^*X$. Let $\omega_X$ be the canonical symplectic form on $T^*X$ and $\mu$ be
a real number. A standard fact says that $\omega_\mu:=\omega_X+2\mu\,\omega_K$
turns $T^*X$ into a symplectic manifold, hence a Poisson manifold with Poisson
bracket $\{\, ,\,\}_\mu$.
  In this article we exhibit a Poisson realization of the simple real Lie
algebra $\mathfrak {su}(n, n)$ on the Poisson manifold $(T^*X, \{\,
,\,\}_\mu)$, i.e., a Lie algebra homomorphism from $\mathfrak {su}(n, n)$ to
$\left(C^\infty(T^*X, \mathbb R), \{\, ,\,\}_\mu\right)$. Consequently one
obtains the Laplace-Runge-Lenz vector for the classical $\mathrm{U}(1)$-Kepler
problem with level $n$ and magnetic charge $\mu$. Since the
McIntosh-Cisneros-Zwanziger-Kepler problems (MICZ-Kepler Problems) are the
$\mathrm{U}(1)$-Kepler problems with level $2$, the work presented here is a
direct generalization of the work by A. Barut and G. Bornzin [ J. Math. Phys.
$\bf 12$ (1971), 841-843] on the classical dynamic symmetry for the MICZ-Kepler
problems.
