  We consider the Cauchy problem for first order systems. Assuming that the set
of the singular points of the characteristic variety is a smooth manifold and
the characteristic values are real and semi-simple we introduce a new class
which is strictly hyperbolic in the transverse direction to the characteristic
manifold. We prove that if the characteristic manifold is either involutive or
symplectic then these transversally strictly hyperbolic systems are strongly
hyperbolic. On the other hand if the characteristic manifold is neither
involutive nor symplectic transversally strictly hyperbolic systems are much
more involved which is discussed taking an interesting example.
