Realizations of kinetic differential equations

The induced kinetic differential equation of a reaction network endowed with mass action type kinetics is a system of polynomial differential equations. The problem studied here is: Given a polynomial differential equation, is it possible to find a network which induces the equation? If yes, can we find a network with some chemically relevant properties (implying also important dynamic consequences), such as reversibility, weak reversibility, zero deficiency, detailed balancing, complex balancing, mass conservation, etc. The constructive answers presented to a series of questions of the above type are useful when fitting a differential equation to measurements, or when trying to find out the dynamic behavior of the solutions of a differential equation. It turns out that some of the results can be applied when trying to solve purely mathematical problems, like the existence of positive solutions to polynomial equation.
Comment: 31 pages, 5 figures. The authors started to work on this paper when enjoying the hospitality of the Erwin Schr\"odinger Institut in Vienna as participants of the meeting Advances in Chemical Reaction Network Theory, 15--19 October, 2018 27 pages (second version)