Semi-abelian surfaces and integrable systems (Q2747919)

For complex analytical Hamiltonian systems there are several definitions of complete algebraic integrability. For Adler and van Moerbeke an algebraic completely integrable system is a system such that the generic Lagrangian manifolds can be completed into abelian varieties [\textit{P. van Moerbeke}, Proc. Symp. Pure Math. 49, 107-131 (1989; Zbl 0688.70012)]. A more general definition is given by Mumford. So, according to Mumford, for the algebraically integrable Hamiltonian systems, the Lagrangian manifolds can be completed into extensions of a linear torus \(\mathbb{C}^{*k}\) by an abelian variety [\textit{D. Mumford}, Tata lectures on theta II, Progress in Mathematics, Vol. 43, Birkhäuser, Boston (1984; Zbl 0549.14014)].NEWLINENEWLINENEWLINEThe paper under review is devoted to the proof that the Lagrange's top, the Kirchhoff's top and the Euler-Poinsot's top are algebraically integrable systems in Mumford's sense. More concretely, it is proved that the Lagrangian manifolds are extensions of \(\mathbb{C}^*\) by an elliptic curve.