  We generalize our previous work on the phase stability and hydrodynamic of
polar liquid crystals possessing local uniaxial $C_{\infty v}$-symmetry to
biaxial systems exhibiting local $C_{2v}$-symmetry. Our work is motivated by
the recently discovered examples of thermotropic biaxial nematic liquid
crystals comprising bent-core mesogens, whose molecular structure is
characterized by a non-polar body axis $({\bf{n}})$ as well as a polar axis
$({\bf{p}})$ along the bisector of the bent mesogenic core which is coincident
with a large, transverse dipole moment. The free energy for this system differs
from that of biaxial nematic liquid crystals in that it contains terms
violating the ${\bf{p}}\to -{\bf{p}}$ symmetry. We show that, in spite of a
general splay instability associated with these parity-odd terms, a uniform
polarized biaxial state can be stable in a range of parameters. We then derive
the hydrodynamic equations of the system, via the Poisson-bracket formalism, in
the polarized state and comment on the structure of the corresponding linear
hydrodynamic modes. In our Poisson-bracket derivation, we also compute the
flow-alignment parameters along the three symmetry axes in terms of microscopic
parameters associated with the molecular geometry of the constituent biaxial
mesogens.
