  We introduce shape fluctuations in a liquid-crystalline system by considering
an elementary Maier--Saupe lattice model for a mixture of uniaxial and biaxial
molecules. Shape variables are treated in the annealed (thermalized) limit. We
analyze the thermodynamic properties of this system in terms of temperature
$T$, concentration $c$ of intrinsically biaxial molecules, and a parameter
$\Delta$ associated with the degree of biaxiality of the molecules. At the
mean-field level, we use standard techniques of statistical mechanics to draw
global phase diagrams, which are shown to display a rich structure, including
uniaxial and biaxial nematic phases, a reentrant ordered region, and many
distinct multicritical points. Also, we use the formalism to write an expansion
of the free energy in order to make contact with the Landau--de Gennes theory
of nematic phase transitions.
