  We discuss counterexamples to the validity of the weak Maximum Principle for
linear elliptic systems with zero and first order couplings and prove, through
a suitable reduction to a nonlinear scalar equation, a quite general result
showing that some algebraic condition on the structure of gradient couplings
and a cooperativity condition on the matrix of zero order couplings guarantee
the existence of invariant cones in the sense of Weinberger [21].
