Integrable Hamiltonian systems on the symplectic realizations of $\textbf{e}(3)^*$

Abstract: 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.