We consider Poiseuille flow of polymeric liquid crystals corresponding to large values of the velocity gradient. The model employed [Ericksen, 1991] proposes governing equations for the velocity field, v, the pressure p, the director n, and the order parameter s. The constitutive functions for the Leslie coefficients alphai derived from the molecular theory of Doi [1981] play a crucial role in the modelling. In addition to the Ericksen number, E, the present model exhibits a new non-dimensional parameter I, that represents the contribution of the elastic free energy of non-gradient type with respect to Frank-Oseen's elasticity. One of the goals of the analysis was to examine the role of s in describing singularities as well as in obtaining regimes which are not predicted by the previous Leslie-Ericksen model. In particular, solutions are obtained that correspond to domain structures parallel to the flow. Such domains are separated by singular lines across which the director experiences jumps of, approximately, 45 degrees with respect to the flow direction. A condition on the size of I is required in order to support such layered structures. The contribution of the energy associated with I turns out to play the role of an elastic surface energy which is, otherwise, neglected in the present model.