Before revision:  One particularly interesting representation of these constraints can be established in terms of a class of monotonous functions which turn out to be critical to conclude uniqueness of equilibrium points in a class of deficiency one networks. One main consequence of such representation is the possibility of a simple constructive proof of the  deficiency one theorem.
After revision:  One particularly interesting representation of these constraints can be established in terms of a class of monotonous functions which turn out to be critical to conclude uniqueness of equilibrium points in a class of positive deficiency networks. The latter result might support constructing an alternative proof of the deficiency one theorem.