  We consider hyperbolic system with nonlinear viscosity such that the
viscosity matrix $B(u)$ is commutating with $A(u)$ the matrix associated to the
convective term. The drift matrix is assumed to be Temple class. First, we
prove the global existence of smooth solutions for initial data with small
total variation. We show that the solution to the parabolic equation converges
to a semi-group solution of the hyperbolic system as viscosity goes to zero.
Furthermore, we prove that the zero diffusion limit coincides with the one
obtained in [Bianchini and Bressan, Indiana Univ. Math. J. 2000].
