The objective of the paper is the pole structure study of solutions in a module theoretic framework, employing the notions of pole module and zero module of a linear transfer function. The paper supplies a complete description of the pole structure. The basic result in this setting is that there is an ``essential'' pole structure which appears in every solution of \(T(\tau)=H(\tau)\) G(\(\tau)\) and which is representable by means of a suitable module determined by T(\(\tau)\) and G(\(\tau)\). Using this module theoretic characterization one cannot only check the presence in the solutions of poles in a certain region, but one can also compute explicitly the list of multiplicities which form the structure. The results concerning the existence of solutions with specific polar properties are effectively improved. The methods proposed in the paper are algebraic and module theoretic and the main tools are represented by the modules of the poles and of the zeros associated with a transfer function. The main result is a complete description of the ``essential'' pole structure which is common to all the solutions H(z). The paper is well organized and written and highly suggested to those involved in the corresponding area of interest.