  In this paper, we study singularities of the Lagrangian fibration given by a
completely integrable system. We prove that a non-degenerate singular fibre
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 neighbourhood of such a fibre is
preserved after any such perturbation. 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
$C^\infty$ smooth integrable perturbations.
