Instability of the Cauchy-Kovalevskaya solution for a class of nonlinear systems

We prove that in any $C^{\infty}$-neighborhood of an analytic Cauchy datum, there exists a smooth function such that the corresponding initial value problem does not have any classical solution for a class of first-order nonlinear systems. We use a method initiated by G.~M\'etivier for elliptic systems based on the representation of solutions and on the FBI transform; in our case the system can be hyperbolic at initial time, but the characteristic roots leave the real line at positive times.