  Extremal black branes are of interest because they correspond to the ground
states of field theories at finite charge density in gauge/gravity duality. The
geometry of such a brane need not be translationally invariant in the spatial
directions along which it extends. A less restrictive requirement is that of
homogeneity, which still allows points along the spatial directions to be
related to each other by symmetries. In this paper, we find large new classes
of homogeneous but anisotropic extremal black brane horizons, which could
naturally arise in gauge/gravity dual pairs. In 4+1 dimensional spacetime, we
show that such homogeneous black brane solutions are classified by the Bianchi
classification, which is well known in the study of cosmology, and fall into
nine classes. In a system of Einstein gravity with negative cosmological term
coupled to one or two massive Abelian gauge fields, we find solutions with an
additional scaling symmetry, which could correspond to the near-horizon
geometries of such extremal black branes. These solutions realize many of the
Bianchi classes. In one case, we construct the complete extremal solution which
asymptotes to AdS space.
