  We study the space spanned by the integer shifts of a bivariate Gaussian
function and the problem of reconstructing any function in that space from
samples scattered across the plane. We identify a large class of lattices, or
more generally semi-regular sampling patterns spread along parallel lines, that
lead to stable reconstruction while having densities close to the critical
value given by Landau's limit. At the critical density, we construct examples
of sampling patterns for which reconstruction fails.
  In the same vein, we also investigate continuous sampling along non-uniformly
scattered families of parallel lines and identify the threshold density of line
configurations at which reconstruction is possible. In a remarkable contrast
with Paley-Wiener spaces, the results are completely different for lines with
rational or irrational slopes.
  Finally, we apply the sampling results to Gabor systems with bivariate
Gaussian windows. As a main contribution, we provide a large list of new
examples of Gabor frames with non-complex lattices having volume close to 1.
