We consider the Cauchy problem for first-order systems. Assuming that the characteristic variety is a smooth manifold and the characteristic values are real and semisimple we introduce a new class which is strictly hyperbolic in the directions transverse to the characteristic manifold. If the propagation cone and the characteristic manifold are compatible we prove that transversally strictly hyperbolic systems are strongly hyperbolic systems. On the other hand if the propagation cone is incompatible with the characteristic manifold then transversally strictly hyperbolic systems are much more involved. In this talk we mainly discuss this case taking an interesting example proposed by G.M¥'etivier.