  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.
