  In this paper we address the issues of absolute continuity for the center
foliation (as well as the disintegration on the non-absolute continuous case)
and rigidity of volume preserving partially hyperbolic diffeomorphisms isotopic
to a linear Anosov on $\mathbb T^3$. It is shown that the disintegration of
volume on center leaves may be neither atomic nor Lebesgue. It is also obtained
results concerning the atomic disintegration. Moreover, the absolute continuity
of the center foliation does not imply smooth conjugacy with its linearization.
Imposing stronger conditions besides absolute continuity on the center
foliation, smooth conjugacy is obtained.
