Singularities of integrable systems and algebraic curves

TL;DR

This paper explores how singularities in integrable systems relate to the properties of their spectral curves, establishing conditions under which these singularities are non-degenerate and characterizing their types.

Contribution

It proves that nodal spectral curves correspond to non-degenerate singularities in a broad class of integrable systems and provides a method to determine singularity types from spectral curve features.

Findings

Nodal spectral curves imply non-degenerate singularities.

The type of singularity can be deduced from double points on the spectral curve.

Linearization on generalized Jacobians facilitates analysis of singularities.

Abstract

We study the relationship between singularities of finite-dimensional integrable systems and singularities of the corresponding spectral curves. For the large class of integrable systems on matrix polynomials, which is a general framework for various multidimensional spinning tops, as well as Beauville systems, we prove that if the spectral curve is nodal, then all singularities on the corresponding fiber of the system are non-degenerate. We also show that the type of a non-degenerate singularity can be read off from the behavior of double points on the spectral curve under an appropriately defined antiholomorphic involution. Our analysis is based on linearization of integrable flows on generalized Jacobian varieties, as well as on the possibility to express the eigenvalues of an integrable vector field linearized at a singular point in terms of residues of certain meromorphic differentials.