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 1 vanishes. The role of the (anomaly-corrected) partition function is played by χρ -the Schur average of the suitably chosen square partiton ρ; such partitions are well-known to correspond to singular vectors of the Virasoro algebra. Further, non-zero are only Schur averages χµ for such µ that have ρ as their r-core, and superintegrability formula features the value of the skew Schur function χ µ/ρ at special point. The associated topological recursion and Harer-Zagier formula generalizations so far remain obscure.