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 $\Tr X^r$, in pure phase, where the naive partition function $\langle 1 \rangle$ vanishes. The role of the (anomaly-corrected) partition function is played by $\left\langle\chi_\rho\right\rangle$ -- the Schur average of the suitably chosen \textit{square} partiton $\rho$; such partitions are well-known to correspond to singular vectors of the Virasoro algebra. Further, non-zero are only Schur averages $\left\langle \chi_\mu\right\rangle$ for such $\mu$ that have $\rho$ as their $r$-core, and superintegrability formula features the value of the \textit{skew} Schur function $\chi_{\mu/\rho}$ at special point. The associated topological recursion and Harer-Zagier formula generalizations so far remain obscure.