Integrable Hamiltonian Systems on the Symplectic Realizations of \(\textbf{e}(3)^*\)

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 the 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. Using 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 Hamiltonian systems on the above symplectic realizations. In this way, after lifting the integrable cases of a gyrostat with a fixed point and of a heavy top, we obtain a large family of integrable Hamiltonian systems on the phase spaces defined by these symplectic realizations.