  The phase space of a gyrostat with a fixed point and a heavy top is the
Lie-Poisson space $\textbf{e}(3)^*\cong \mathbb{R}^3\times \mathbb{R}^3$ dual
to the Lie algebra $\textbf{e}(3)$ of Euclidean group $E(3)$. One has three
naturally distinguished Poisson submanifolds of $\textbf{e}(3)^*$: (i) the
dense open submanifold $\mathbb{R}^3\times \dot{\mathbb{R}}^3\subset
\textbf{e}(3)^*$ which consists of all $4$-dimensional symplectic leaves
($\vec{\Gamma}^2>0$); (ii) the $5$-dimensional Poisson submanifold of
$\mathbb{R}^3\times \dot{\mathbb{R}}^3$ defined by $\vec{J}\cdot \vec{\Gamma} =
\mu ||\vec{\Gamma}||$; (iii) the $5$-dimensional Poisson submanifold of
$\mathbb{R}^3\times \dot{\mathbb{R}}^3$ defined by $\vec{\Gamma}^2 = \nu^2$,
where $\dot{\mathbb{R}}^3:= \mathbb{R}^3\backslash \{0\}$, $(\vec{J},
\vec{\Gamma})\in \mathbb{R}^3\times \mathbb{R}^3\cong \textbf{e}(3)^*$ and $\nu
< 0 $, $\mu$ are some fixed real parameters. Basing on the $U(2,2)$-invariant
symplectic structure of Penrose twistor space we find full and complete
$E(3)$-equivariant symplectic realizations of these Poisson submanifolds which
are $8$-dimensional for (i) and $6$-dimensional for (ii) and (iii). As a
consequence of the above Hamiltonian systems on $\textbf{e}(3)^*$ lift to the
ones on the above symplectic realizations. In such a way after lifting
integrable cases of gyrostat with a fixed point, as well as of heavy top, we
obtain a large family of integrable Hamiltonian systems on the phase spaces
defined by these symplectic realizations.
