An orderly linear PDE system with analytic initial conditions with a non-analytic solution

We give a linear PDE system, with analytic initial conditions given w.r.t. an orderly ranking, the solution of which is not analytic (moreover the solution is not Gevrey for any order). This example proves that the analyticity Riquier theorem (generalization of the Cauchy–Kovalevskaya theorem) does not generalize to PDE systems endowed with orderly rankings.