The effects of the breaking of an internal symmetry on the singularities of the scattering matrix, especially near thresholds, is discussed in terms of the $K$-matrix formalism. It is shown that, in first order in the symmetry-breaking coupling constant, the $K$ matrix transforms like the symmetry-breaking Hamiltonian. This result provides a justification for the Gell-Mann-Okubo mass formula for narrow resonances with nonzero orbital angular momentum. The position of poles in different sheets of the energy Riemann surface and their displacement with the symmetry-breaking parameter is investigated in detail, particularly for $s$ waves. The specific case of baryon-baryon scattering in the triplet $s\ensuremath{-}d$ states is discussed using a potential model with broken ${\mathrm{SU}}_{3}$ symmetry. On the basis of information from low-energy $np$ scattering and $\ensuremath{\Lambda}p$ final-state interaction we suggest that a resonance probably exists in the $I=1$, $\ensuremath{\Xi}N$ system, below the $\ensuremath{\Sigma}\ensuremath{\Lambda}$ threshold. The Gell-Mann-Okubo mass formula would have placed the resonance close to the $\ensuremath{\Sigma}\ensuremath{\Sigma}$ threshold.