ИСТИНА

An integrable Hamiltonian system with n degrees of freedom is given by n functionally independent functions pairwise in involution on a symplectic 2n-manifold M. Consider the singular Lagrangian fibration on M whose fibres are connected components of the common level sets of the given functions. By a semilocal singularity, we mean the fibration germ at a singular fibre. We give a (weak) sufficient condition for a nondegenerate semilocal singularity of a real-analytic integrable system to be structurally stable under real-analytic integrable perturbations. We also give a symplectic real-analytic classification of these singularities. As an illustration, we show that a saddle-saddle singularity of the Kovalevskaya top is structurally stable under real-analytic integrable perturbations, but structurally unstable under smooth integrable perturbations.