Poisson surfaces and algebraically completely integrable systems

TL;DR

This paper explores a special class of algebraically integrable systems linked to certain ruled surfaces, introducing a new integrable system that deforms the Hitchin system, expanding the understanding of geometric structures in integrable systems.

Contribution

It identifies a previously overlooked ruled surface associated with jet bundles and constructs a new integrable system that deforms the Hitchin system.

Findings

Introduces a new integrable system related to jet bundle ruled surfaces.

Shows this system is a deformation of the Hitchin system.

Expands classification of algebraically integrable systems.

Abstract

One can associate to many of the well known algebraically integrable systems of Jacobians (generalized Hitchin systems, Sklyanin) a ruled surface which encodes much of its geometry. If one looks at the classification of such surfaces, there is one case of a ruled surface that does not seem to be covered. This is the case of projective bundle associated to the first jet bundle of a topologically nontrivial line bundle. We give the integrable system corresponding to this surface; it turns out to be a deformation of the Hitchin system.