Algebraic integrability: the Adler-Van Moerbeke approach (Q691171)

In this paper, an overview of the latest achievements in the area of algebraic completely integrable systems is presented. Following \textit{M. Adler} and {P. van Moerbeke} [Invent. Math. 67, 297--331 (1982; Zbl 0539.58012)], a dynamical system is ``algebraic completely integrable'' if it can be linearized on a complex algebraic torus. In other words, these are integrable systems whose trajectories are straight line motions on abelian varieties (complex algebraic tori). So the flow can be solved by quadrature, that is to say their solutions can be expressed in term of abelian integrals. This work consists of the following basic parts: 1. Algebraic complete integrability; 2. The Liouville-Arnold-Adler-Van Moerbeke theorem; 3. A five-dimensional system; 4. The Hénon-Heiles system; 5. The Kowalewski rigid body motion; 6. The geodesic flow on \(\mathrm{SO}(n)\) for a left invariant metric.