The classical dynamic symmetry for the $\mathrm{S}\mathrm{p}(1)$-Kepler problems

A Poisson realization of the simple real Lie algebra ${\mathfrak{s}\mathfrak{o}}^{\ast }(4n)$ on the phase space of each $\mathrm{S}\mathrm{p}(1)$-Kepler problem is exhibited. As a consequence one obtains the Laplace-Runge-Lenz vector for each classical $\mathrm{S}\mathrm{p}(1)$-Kepler problem. The verification of these Poisson realizations is greatly simplified via an idea due to A. Weinstein. The totality of these Poisson realizations is shown to be equivalent to the canonical Poisson realization of ${\mathfrak{s}\mathfrak{o}}^{\ast }(4n)$ on the Poisson manifold $T^\mathbb H_^n/$\mathrm{S}\mathrm{p}$(1)$\mathrm{.}(Here$\mathbb H_*^n:=\mathbb H^n\backslash {0}$ and the Hamiltonian action of $\mathrm{S}\mathrm{p}(1)$ on $T^{\ast }{\mathbb{H}}_{\ast }^n$ is induced from the natural right action of $\mathrm{S}\mathrm{p}(1)$ on ${\mathbb{H}}_{\ast }^n$. )