  This paper concerns the recent Virasoro conjecture for the theory of stable
pairs on a 3-fold proposed by Oblomkov, Okounkov, Pandharipande and the author
in arXiv:2008.12514. Here we extend the conjecture to 3-folds with
non-$(p,p)$-cohomology and we prove it in two specializations. For the first
specialization, we let $S$ be a simply-connected surface and consider the
moduli space $P_n(S\times \mathbb{P}^1, n[\mathbb{P}^1])$, which happens to be
isomorphic to the Hilbert scheme $S^{[n]}$ of $n$ points on $S$. The Virasoro
constraints for stable pairs, in this case, can be formulated entirely in terms
of descendents in the Hilbert scheme of points. The two main ingredients of the
proof are the toric case and the existence of universal formulas for integrals
of descendents on $S^{[n]}$. The second specialization consists in taking the
3-fold $X$ to be a cubic and the curve class $\beta$ to be the line class. In
this case we compute the full theory of stable pairs using the geometry of the
Fano variety of lines.
