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

On hyperbolic systems with time dependent H\"older characteristics