We have performed mm-wave observations with the IRAM 30m telescope of the 12CO J=2-1 and J=1-0, 13CO J=2-1 and J=1-0, and SiO J=5-4 transitions in the symbiotic stars R Aqr, CH Cyg, and HM Sge. The data were analyzed by means of a simple analytical description of the general properties of molecular emission from the inner shells around the cool star. Numerical calculations of the expected line profiles, taking into account the level population and radiative transfer under such conditions, were also performed. Weak emission of 12CO J=2-1 and J=1-0 was detected in R Aqr and CH Cyg; a good line profile of 12CO J=2-1 in R Aqr was obtained. The intensities and profile shapes of the detected lines are compatible with emission coming from a very small shell around the Mira-type star, with a radius comparable to or slightly smaller than the distance to the hot dwarf companion, 10$^{14}$ - 2 10$^{14}$ cm. We argue that other possible explanations are improbable. This region probably shows properties similar to those characteristic of the inner shells around standard AGB stars: outwards expansion at about 5 - 25 km/s, with a significant acceleration of the gas, temperatures decreasing with radius between about 1000 and 500 K, and densities ~ 10$^9$ - 3 10$^8$ cm$^{-3}$. Our model calculations are able to explain the asymmetric line shape observed in 12CO J=2-1 from R Aqr, in which the relatively weaker red part of the profile would result from selfabsorption by the outer layers (in the presence of a velocity increase and a temperature decrease with radius). The mass-loss rates are somewhat larger than in standard AGB stars, as often happens for symbiotic systems. In R Aqr, we find that the total mass of the CO emitting region is ~ 2 - 3 10$^{-5}$ Mo, corresponding to M' ~ 5 10$^{-6}$ - 10$^{-5}$ Mo/yr, and compatible with results obtained from dust emission. Taking into account other existing data on molecular emission, we suggest that the small extent of the molecule-rich gas in symbiotic systems is mainly due to molecule photodissociation by the radiation of the hot dwarf star.
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