The "Null-A" superintegrability for monomial matrix models

We find that superintegrability (character expansion) property persists in the exotic sector of the monomial non-Gaussian matrix model, with potential $\text{Tr}{X}^r$, in pure phase, where the naive partition function $\langle 1 \rangle$ vanishes. The role of the (anomaly-corrected) partition function is played by $⟨{\chi }_{\rho }⟩$ – the Schur average of the suitably chosen $\mathit{\text{square}}$ partiton $\rho$; such partitions are well-known to correspond to singular vectors of the Virasoro algebra. Further, non-zero are only Schur averages $⟨{\chi }_{\mu }⟩$ for such $\mu$ that have $\rho$ as their $r$-core, and superintegrability formula features the value of the $\mathit{\text{skew}}$ Schur function ${\chi }_{\mu \mathrm{/}\rho }$ at special point. The associated topological recursion and Harer-Zagier formula generalizations so far remain obscure.