  We continue our study of initial-value problems for fully nonlinear systems
exhibiting strong or weak defects of hyperbolicity. We prove that, regardless
of the initial Sobolev regularity, the initial-value problem has no local $H^s$
solutions, for $s > s_0 + d/2,$ if the principal symbol has a strong, or even
weak, defect of hyperbolicity, and the purely imaginary eigenvalues of the
principal symbol are semi-simple and have constant multiplicity. The index $s_0
> 0$ depends on the severity of the defect of hyperbolicity. These results
recover and extend previous work from G. M\'etivier [{\it Remarks on the well
posedness of the nonlinear Cauchy problem,} 2005], N.Lerner, Y. Morimoto, C.-J.
Xu [{\it Instability of the Cauchy-Kovalevskaya solution for a class of
non-linear systems}, 2010] and N. Lerner, T. Nguyen, B. Texier, {\it The onset
of instability in first-order systems}, 2018]
