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We prove that, for first-order, fully nonlinear systems of partialdifferential equations, under an hypothesis of ellipticity for the principalsymbol, the Cauchy problem has no solution within a range of Sobolev indicesdepending on the regularity of the initial datum. This gives a new and greatlydetailed proof of a result of G. M\'etivier [{\it Remarks on the Cauchyproblem}, 2005]. We then extend this result to systems experiencing atransition from hyperbolicity and ellipticity, in the spirit of recent work byN. Lerner, Y. Morimoto, and C.-J. Xu, [{\it Instability of theCauchy-Kovalevskaya solution for a class of non-linear systems}, 2010], and N.Lerner, T. Nguyen and B. Texier [{\it The onset of instability in first-ordersystems}, 2018].

On M\'etivier's Lax-Mizohata theorem and extensions to weak defects of hyperbolicity. Part one