Algebraic geometry and stability for integrable systems

•We present an algebro-geometric method of stability analysis for integrable systems.•The method allows to conclude about stability from the shape of the spectral curve.•We solve the stability problem for the free nn-dimensional rigid body as an example.

In 1970s, a method was developed for integration of nonlinear equations by means of algebraic geometry. Starting from a Lax representation with spectral parameter, the algebro-geometric method allows to solve the system explicitly in terms of theta functions of Riemann surfaces. However, the explicit formulas obtained in this way fail to answer qualitative questions such as whether a given singular solution is stable or not. In the present paper, the problem of stability for equilibrium points is considered, and it is shown that this problem can also be approached by means of algebraic geometry.