We have used theology, rheo-optical observations, and small-angle neutron scattering under shear to investigate the flow, orientation, and texture properties of nematic solutions of equilibrium or "living" polymers. The elementary objects of these liquid crystalline phases are wormlike micelles, that is, flexible cylindrical aggregates made of amphiphile molecules. The aqueous surfactant solutions investigated here (cetylpyridinium chloride and hexanol in 0.2 M NaCl-brine) exhibit the classical structural sequence : isotropic/nematic/hexagonal with increasing surfactant concentration phi. The nematic phase observed for phi = 34%-38% consists of very long cylindrical micelles closely packed parallel to each other, thus producing long-range orientational order. We have shown that when subjected to a steady shear, the nematic phase of living polymers behaves very similarly to that of the conventional liquid crystalline polymers. From the rheo-optical observations, we demonstrated that so-called band textures show up perpendicular to the flow direction after cessation of the shearing, indicating a modulation of the director orientation. As for Liquid crystalline polymers, these textures are interpreted in terms of relaxation of the Frank distorsional energy accumulated during the now. The time evolution of the average band spacing of the texture could be determined and compared to recent theoretical predictions;e.g. in the low-shear regime the squared band spacing scales linearly with time. We also study the evolution of the order parameter of the nematic state (actually spatially averaged over the mesoscopic textures observed during the flow or at rest) determined from small-angle neutron scattering spectra obtained under shear. As remarkable features, we found out that above a characteristic shear rate, gamma(c), the nematic living polymers form an orientational monodomain which in addition is remarkably stable over a long period of time. The present data are compared to those reported for liquid crystalline polymers.