  We show that there exist complete and minimal systems of time-frequency
shifts of Gaussians in $L^2(\mathbb{R})$ which are not strong Markushevich
basis (do not admit the spectral synthesis). In particular, it implies that
there is no linear summation method for general Gaussian Gabor expansions. On
the other hand we prove that the spectral synthesis for such Gabor systems
holds up to one dimensional defect.
