We prove results on existence and uniqueness of solutions of the Cauchy Problem for $2\times 2$ weakly hyperbolic systems. The results follow from an extension for systems of the work by O. Oleinik done in the scalar case.
"On the Cauchy problem for $2\times 2$ weakly hyperbolic systems." Osaka J. Math. 42 (4) 831 - 860, December 2005.