Integrable Systems

8Generalized Algebraic Completely Integrable Systems

Some interesting cases of integrable systems, to be discussed in this chapter, appear as coverings of algebraic completely integrable systems. The manifolds in variant by the complex flows are coverings of Abelian varieties and these systems are called generalized algebraic completely integrable. The latter are completely integrable in the sense of Arnold–Liouville and so generically, the compact connected manifolds invariant by the real flows are tori, the real parts of complex affine coverings of Abelian varieties. Also, we will see how some algebraic completely integrable systems can be constructed from known algebraic completely integrable in the generalized sense. We will see that a large class of algebraic completely integrable systems in the generalized sense are part of new algebraic completely integrable systems. We consider (as examples of applications) the Hénon–Heiles problem, the Ramani–Dorizzi–Grammaticos (RDG) potential, the Yang–Mills system, Goryachev–Chaplygin and Lagrange tops, as well as other interesting systems related to these examples.

8.1. Generalities

There are many examples of differential equations ż = f(z), z ∈ ℂ^m, which have the weak Painlevé property that all movable singularities of the general solution have only a finite number of branches, and some integrable systems appear as coverings of algebraic completely integrable systems. The manifold invariants by the complex flows are coverings ...