Generalized algebraic completely integrable systems

Abstract

We tackle in this paper the study of generalized algebraic completely integrable systems. Some interesting cases of integrable systems appear as coverings of algebraic completely integrable systems. The manifolds invariant by the complex flows are coverings of Abelian varieties and these systems are called algebraic completely integrable in the generalized sense. The later are completely integrable in the sense of Arnold-Liouville. We shall see how some algebraic completely integrable systems can be constructed from known algebraic completely integrable in the generalized sense. A large class of algebraic completely integrable systems in the generalized sense, are part of new algebraic completely integrable systems. We discuss some interesting and well known examples : a 4-dimensional algebraically integrable system in the generalized sense as part of a 5-dimensional algebraically integrable system, the Hénon-Heiles and a 5-dimensional system, the RDG potential and a 5-dimensional system, the Goryachev-Chaplygin top and a 7-dimensional system, the Lagrange top, the (generalized) Yang-Mills system and cyclic covering of Abelian varieties.

Keywords

• integrable systems
• abelian varieties
• surfaces of general type