It is well-known that the majority of finite-dimensional integrable systems can be solved by linearizing them on the Jacobian of the associated spectral curve. In the talk, I will explain how to extend this technique to study singularities of integrable systems.

Singularities of integrable systems are those points where the first integrals become dependent. In the first part of the talk, I will discuss some aspects of the local theory of such singularities and show how to use generalized Jacobians to prove that singularities corresponding to nodal spectral curves are non-degenerate.

Then, time permitting, I will discuss the global structure of singular fibers corresponding to reducible spectral curves. Such fibers turn out to have interesting combinatorics related to convex polytopes, orientations of graphs etc. I will also mention the conjectural relation of these fibers to certain compactifications of generalized Jacobians.