It is well known that the rigid rod model for the dynamics of nematic polymers can be formulated in terms of a non-linear partial differential equation for the orientational distribution function, the so-called Smoluchowski Equation, and the rheological predictions are then obtained by integrating this equation numerically. The integration requires cumbersome numerical schemes even for very simple flow fields, and thus techniques which reduce the complexity of the problem by transforming the partial differential to a set of a few ordinary differential equations are attractive. In these approximations, the bifurcation structure of the original model should be maintained, however. In this work, we have determined for the first time the bifurcation structure of the Smoluchowski Equation; the rigid rod model is exhaustively studied for the case of a shear flow.