  We present an anisotropic cosmological model based on a new exact solution of
Einstein equations. The matter content consists of an anisotropic scalar field
minimally coupled to gravity and of two isotropic perfect fluids that represent
dust matter and radiation. The spacetime is described by a spatially
homogeneous, Bianchi type III metric with a conformal expansion. The model
respects the evolution of the scale factor predicted by standard cosmology, as
well as the isotropy of the cosmic microwave background. Remarkably, the
introduction of the scalar field, apart from explaining the spacetime
anisotropy, gives rise to an energy density that is close to the critical
density. As a consequence, the model is quasiflat during the entire history of
the universe. Using these results, we are also able to construct approximate
solutions for shear-free cosmological models with rotation. We finally carry
out a quantitative discussion of the validity of such solutions, showing that
our approximations are acceptably good if the angular velocity of the universe
is within the observational bounds derived from rotation of galaxies.
