  We use the concept of reproducing pairs to study Gabor systems at critical
density. First, we present a generalization of the Balian-Low theorem to the
reproducing pairs setting. Then, we prove our main result that there exists a
reproducing partner for the Gabor system of integer time-frequency shifts of
the Gaussian. In other words, the coefficients for this Gabor expansion of a
square integrable function can be calculated using inner products with an
unstructured family of vectors in $ L^2(\R)$. This solves the possibly last
open question for this system.
