  How often does it occur that the measure of maximal entropy of a system is an
SRB measure? We study this question for $C^{1+\alpha}$ partially hyperbolic
diffeomorphisms isotopic to Anosov (DA-diffeomorphisms) on ${\mathbb T}^{3}$,
and establish a rigidity result: the measure of maximal entropy is an SRB
measure if and only if the sum of its positive Lyapunov exponents coincides
with that of the linear Anosov map $A$ on all periodic orbits of the support of
the measure. In that case, the measure is also the unique physical measure.\par
We also show non-Anosov examples satisfying this condition, both in the
conservative and in the non-conservative setting. \par Finally, we prove that a
volume-preserving $C^{1+\alpha}$ DA-diffeomorphism on ${\mathbb T}^{3}$ is
Anosov if all Lyapunov exponents coincide almost everywhere with those of the
linear Anosov in the isotopy class. Consequently, a smooth DA-diffeomorphism is
smoothly conjugated to its linear part if and only if all Lyapunov exponents
coincide almost everywhere with those of its linear part.
