I will discuss firs the validity of the weak Maximum Principle (wMP) for vector functions u = (u1, .., um) satisfying systems of the form Au + Cu ≥ 0 in a bounded open set Ω of Rn where A is a diagonal matrix of linear degenerate second order elliptic operators and C is a cooperative matrix. Next some counterexamples to the validity of (wMP) are discussed when non diagonal couplings in first order partial derivatives of the ui appear in the system. In this more general setting I will show, through a suitable reduction to a nonlinear scalar equation of Bellman type, 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.