It is well-known that regular fibers of most finite-dimensional integrable systems can be described as open subsets in Abelian varieties. The general question I am going to address is how to generalize this description to singular fibers.

I will concentrate on the integrable system on matrix polynomials. The fibers of this system are enumerated by plane algebraic curves, and each regular fiber is (modulo the action of some gauge group) an open subset in the Jacobian of the corresponding curve. I will explain what happens when one considers a fiber corresponding to a singular curve. Such fibers turn out to have interesting combinatorics related to convex polytopes, orientations of graphs etc.

The talk will be mostly combinatorial. No seriuos background in algebraic geometry is assumed.