  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.
