Structurally Stable Nondegenerate Singularities of Integrable Systems

Abstract

In this paper, we study singularities of the Lagrangian fibration given by a completely integrable system. We prove that a nondegenerate singular fiber satisfying the so-called connectedness condition is structurally stable under (small enough) real-analytic integrable perturbations of the system. In other words, the topology of the fibration in a neighborhood of such a fiber is preserved after any such perturbation. As an illustration, we show that a simple saddle-saddle singularity of the Kovalevskaya top is structurally stable under real-analytic integrable perturbations and not structurally stable under \(C^\infty\)-smooth integrable perturbations.