  We consider a Piecewise Deterministic Markov Process given by random
switching between finitely many vector fields vanishing at $0$. It has been
shown recently that the behaviour of this process is mainly determined by the
signs of Lyapunov exponents. However, results have only been given when all
these exponents have the same sign. In this note, we consider the degenerate
case where the process leaves invariant some face and results are stated when
the Lyapunov exponents are of opposite signs. Applications are given to Lorenz
vector fields with switching, and to SIRS model in random environment.
