  We study hyperbolic systems with multiplicities and smo\-oth coefficients. In
the case of non-analytic, smooth coefficients, we prove well-posedness in any
Gevrey class and when the coefficients are analytic, we prove $C^\infty$
well-posedness. The proof is based on a transformation to block Sylvester form
introduced by D'Ancona and Spagnolo in Ref. 9 which increases the system size
but does not change the eigenvalues. This reduction introduces lower order
terms for which appropriate Levi-type conditions are found. These translate
then into conditions on the original coefficient matrix. This paper can be
considered as a generalisation of Ref. 12, where weakly hyperbolic higher order
equations with lower order terms were considered.
