  In this paper we analyze the tangential symmetries of Darboux integrable
decomposable exterior differential systems. The decomposable systems generalize
the notion of a hyperbolic exterior differential system and include the classic
notion of Darboux integrability for first order systems and second order scalar
equations. For Darboux integrable systems the general solution can be found by
integration (solving ordinary differential equations). We show that this
property holds for our generalized systems as well.
  We give a geometric construction of the Lie algebras of tangential symmetries
associated to the Darboux integrable systems. This construction has the
advantage over previous constructions that our construction does not require
the use of adapted coordinates and works for arbitrary dimension of the
underlying manifold. In particular it works for the prolongations of
decomposable exterior differential systems.
