  We examine the behaviour of homogeneous, anisotropic space-times,
specifically the locally rotationally symmetric Bianchi types $I$ and $VII_o$
in the presence of anisotropic matter. By finding an appropriate constant of
the motion, and transforming the equations of motion we are able to provide
exact solutions in the presence perfect fluids with anisotropic pressures. The
solution space covers matter consisting of a single perfect fluid which
satisfies the weak energy condition and is rich enough to contain solutions
which exhibit behaviour which is qualitatively distinct from the isotropic
sector. Thus we find that there is more `matter that matters' close to a
homogeneous singularity than the usual stiff fluid. Example metrics are given
for cosmologies whose matter sources are magnetic fields, relativistic
particles, cosmic strings and domain walls.
