Observations of 28SiO v=0 J=1-0 line emission (7-mm wavelength) from AGB stars show in some cases peculiar profiles, composed of a central intense component plus a wider plateau. Very similar profiles have been observed in CO lines from some AGB stars and most post-AGB nebulae and, in these cases, they are clearly associated with the presence of conspicuous axial symmetry and bipolar dynamics. We present systematic observations of 28SiO v=0 J=1-0 emission in 28 evolved stars, performed with the 40~m radio telescope of the IGN in Yebes, Spain. We find that the composite core plus plateau profiles are almost always present in O-rich Miras, OH/IR stars, and red supergiants. They are also found in one S-type Mira ($chi$ Cyg), as well as in two semiregular variables (X Her and RS Cnc) that are known to show axial symmetry. In the other objects, the profiles are simpler and similar to those of other molecular lines. The composite structure appears in the objects in which SiO emission is thought to come from the very inner circumstellar layers, prior to dust formation. The central spectral feature is found to be systematically composed of a number of narrow spikes, except for X Her and RS Cnc, in which it shows a smooth shape that is very similar to that observed in CO emission. These spikes show a significant (and mostly chaotic) time variation, while in all cases the smooth components remain constant within the uncertainties. The profile shape could come from the superposition of standard wide profiles and a group of weak maser spikes. Alternatively, we speculate that the very similar profiles detected in objects that are axisymmetric may be indicative of the systematic presence of a significant axial symmetry in the very inner circumstellar shells around AGB stars; the presence of such symmetry would be independent of the probable weak maser effects in the central spikes.
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