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Singular Gelfand-Tsetlin modules of index 2 are modules whose tableaux basesmay have singular pairs but no singular triples of entries on each row. In thispaper we construct singular Gelfand-Tsetlin modules for arbitrary singularcharacter of index 2. Explicit bases of derivative tableaux and the action ofthe generators of $\mathfrak{gl}(n)$ are given for these modules. Ourconstruction leads to new families of irreducible Gelfand-Tsetlin modules andalso provides tableaux bases for some simple Verma modules.

New singular Gelfand-Tsetlin $\mathfrak{gl} (n)$-modules of index 2