  We develop the complete theory for the collective plasmon modes of an
interacting electron system in the presence of explicit mass (or velocity)
anisotropy in the corresponding non-interacting situation, with the effective
Fermi velocity being different along different axes. Such effective mass
anisotropy is common in solid state materials (e.g., silicon or germanium),
where the Fermi surface is often not spherical. We find that the plasmon
dispersion itself develops significant anisotropy in such systems, and the
commonly used isotropic approximation of using a density of states or optical
effective mass does not work for the anisotropic system. We predict a
qualitatively new phenomenon in anisotropic systems with no corresponding
isotropic analog, where the plasmon mode along one direction decays into
electron-hole pairs through Landau damping while the mode remains undamped and
stable along a different directions
