  In this paper we establish in the fast diffusion range the higher
integrability of the spatial gradient of weak solutions to porous medium
systems. The result comes along with an explicit reverse H\"older inequality
for the gradient. The novel feature in the proof is a suitable intrinsic
scaling for space-time cylinders combined with reverse H\"older inequalities
and a Vitali covering argument within this geometry. The main result holds for
the natural range of parameters suggested by other regularity results. Our
result applies to general fast diffusion systems and includes both, nonnegative
and signed solutions in the case of equations. The methods of proof are purely
vectorial in their structure.
