  In this paper we study higher order weakly hyperbolic equations with time
dependent non-regular coefficients. The non-regularity here means less than
H\"older, namely bounded coefficients. As for second order equations in
\cite{GR:14} we prove that such equations admit a `very weak solution' adapted
to the type of solutions that exist for regular coefficients. The main idea in
the construction of a very weak solution is the regularisation of the
coefficients via convolution with a mollifier and a qualitative analysis of the
corresponding family of classical solutions depending on the regularising
parameter. Classical solutions are recovered as limit of very weak solutions.
Finally, by using a reduction to block Sylvester form we conclude that any
first order hyperbolic system with non-regular coefficients is solvable in the
very weak sense.
