  We prove the existence of self-similar fundamental solutions (SSF) of the
anisotropic porous medium equation in the suitable fast diffusion range. Each
of such SSF solutions is uniquely determined by its mass. We also obtain the
asymptotic behaviour of all finite-mass solutions in terms of the family of
self-similar fundamental solutions. Time decay rates are derived as well as
other properties of the solutions, like quantitative boundedness, positivity
and regularity. The combination of self-similarity and anisotropy is essential
in our analysis and creates serious mathematical difficulties that are
addressed by means of novel methods.
