  In this paper we study the well-posedness of weakly hyperbolic systems with
time dependent coefficients. We assume that the eigenvalues are low regular, in
the sense that they are H\"older with respect to $t$. In the past these kind of
systems have been investigated by Yuzawa \cite{Yu:05} and Kajitani \cite{KY:06}
by employing semigroup techniques (Tanabe-Sobolevski method). Here, under a
certain uniform property of the eigenvalues, we improve the Gevrey
well-posedness result of \cite{Yu:05} and we obtain well-posedness in spaces of
ultradistributions as well. Our main idea is a reduction of the system to block
Sylvester form and then the formulation of suitable energy estimates inspired
by the treatment of scalar equations in \cite{GR:11}
