Spectral representations and description of a superconducting state with $\mathbf{S}$-Type order parameter $\Delta(\mathbf{k})$

The inclusion of a singular contribution to the spectral intensity of the anomalous correlation function is shown to regain the sum rule and remove the unjustified forbidding of the $S$-symmetry order parameter in superconductors with strong correlations. For the order parameter of this symmetry, the solution to the self-consistency equation is analyzed beyond the nearest-neighbor approximation.