The interaction between flow and orientation of liquid crystalline polymers (LCPs) creates remarkable heterogeneous patterns in which defects, or singular solutions, serve to mediate a confluence of ordered nematic phases. The origin of defects remains a mystery. It is therefore valuable to have models for LCP flows that provide some evidence of defects, and of the corresponding physical competition between flow and LCP properties. In this direction, the flow-orientation moment-averaged Doi model is studied with an imposed elongational flow. Nonhomogeneous, biaxial nematic patterns are discovered in both axial and planar elongation. These exact solutions consist of spatially varying directors in the plane orthogonal to the flow axis, coupled with homogeneous biaxial order parameter equilibria fixed by the LCP concentration (N) and elongation rate (nu). For each (N, nu), the following patterns coexist all with identical order parameter values: the homogeneous patterns of Rey [Macromol. Theory Simul. 4, 857-872 (1995)]; radially symmetric director patterns; and finally, director patterns periodic in the cylindrical azimuthal angle. The nonhomogeneous structures are distinguished by the presence of core defects along the axis of flow symmetry, characterized by a logarithmic pressure singularity at the core.