There is a group of binary post-AGB stars that show a conspicuous NIR excess, usually assumed to arise from hot dust in very compact possibly rotating disks. These stars are surrounded by significantly fainter nebulae than the standard, well studied protoplanetary and planetary nebulae (PPNe, PNe). We present high-sensitivity mm-wave observations of CO lines in 24 objects of this type. CO emission is detected in most observed sources and the line profiles show that the emissions very probably come from disks in rotation. We derive typical values of the disk mass between 1e-3 and 1e-2 Mo, about two orders of magnitude smaller than the (total) masses of standard PPNe. The high-detection rate (upper limits being in fact not very significant) clearly confirm that the NIR excess of these stars arises from compact disks in rotation, very probably the inner parts of those found here. Low-velocity outflows are also found in about eight objects, with moderate expansion velocities of ~ 10 km/s, to be compared with the velocities of about 100 km/s often found in standard PPNe. Except for two sources with complex profiles, the outflowing gas in our objects represents a minor nebular component. Our simple estimates of the disk typical sizes yields values ~ 0.5 - 1 arcsec, i.e. between 5e15 and 3e16 cm. Estimates of the linear momenta carried by the outflows, which can only be performed in a few well studied objects, also yield moderate values, compared with the linear momenta that can be released by the stellar radiation pressure (contrary, again, to the case of the very massive and fast bipolar outflows in standard PPNe, that are strongly overluminous). The mass and dynamics of nebulae around various classes of post-AGB stars differ very significantly, and we can expect the formation of PNe with very different properties.
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