  A Gabor system in $L^2(\mathbb{R})$, generated by a window $g\in
L^2(\mathbb{R})$ and associated with a sequence of times and frequencies
$\Gamma\subset\mathbb{R}^2$, is a set formed by translations in time and
modulations of $g$. In this paper we consider the case when $g$ is the Gaussian
function and $\Gamma$ is a sequence whose associated Gabor system
$\mathcal{G}_\Gamma$ is complete and minimal in $L^2(\mathbb{R})$. We consider
two main cases: that of the lattice without one point and that of the sequence
constructed by Ascensi, Lyubarskii and Seip lying on the union of the
coordinate axes of the time-frequency space. We study the stability problem for
these two systems. More precisely, we describe the perturbations of $\Gamma$
such that the associated Gabor systems remain to be complete and minimal. Our
method of proof is based essentially on estimates of some infinite products.
