Abstract: In this paper, starting from a comparison of two optical symmetrizing systems, a detailed analysis of the symmetrization of astigmatic Hermite-Gaussian beams and related problems is made by using the generalized HuygensFresnel diffraction integral .It is shown that the symmetrizing system plays a role in transforming astigmatic beams into symmetric beams with real argument and eliminating the xy coupling term. Thus, they can retain their shape unchanged upon further free space propagation. Nevertheless, the symmetric complex beams with the xy coupling term do not possess this propagation invariance.