Integrable Hamiltonian Systems on the Symplectic Realizations of e(3)

The phase space of a gyrostat with a fixed point and a heavy top is the Lie-Poisson space e(3)* congruent to R-3 x R-3 dual to the Lie algebra e(3) of the Euclidean group E(3). One has three naturally distinguished Poisson submanifolds of e(3)* : (i) the dense open submanifold R-3 x R-3 subset of e(3)* which consists of all 4-dimensional symplectic leaves ((Gamma) over right arrow2 > 0); (ii) the 5-dimensional Poisson submanifold of R3 x. R 3 defined by (J) over right arrow center dot (Gamma) over right arrow = mu||(Gamma) over right arrow||; (iii) the 5-dimensional Poisson submanifold of R-3 x R (3) defined by Gamma(2) =upsilon(2), where. R-3 := R-3\{0}, (J, Gamma) is an element of R-3 xR(3) congruent to 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 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.