  We construct a Poisson map $\mathbf{M}\colon T^{*}\mathbb{C}^{2} \to
\mathfrak{se}(3)^{*}$ with respect to the canonical Poisson bracket on
$T^{*}\mathbb{C}^{2} \cong T^{*}\mathbb{R}^{4}$ and the $(-)$-Lie--Poisson
bracket on the dual $\mathfrak{se}(3)^{*}$ of the Lie algebra of the special
Euclidean group $\mathsf{SE}(3)$. The essential part of this map is the
momentum map associated with the cotangent lift of the natural right action of
the semidirect product Lie group $\mathsf{SU}(2) \ltimes \mathbb{C}^{2}$ on
$\mathbb{C}^{2}$. This Poisson map gives rise to a canonical Hamiltonian system
on $T^{*}\mathbb{C}^{2}$ whose solutions are mapped by $\mathbf{M}$ to
solutions of the heavy top equations. We show that the Casimirs of the heavy
top dynamics and the additional conserved quantity of the Lagrange top
correspond to the Noether conserved quantities associated with certain
symmetries of the canonical Hamiltonian system. We also construct a
Lie--Poisson integrator for the heavy top dynamics by combining the Poisson map
$\mathbf{M}$ with a simple symplectic integrator, and demonstrate that the
integrator exhibits either exact or near conservation of the conserved
quantities of the Kovalevskaya top.
