The microstructural relaxation equation for a symmetric, traceless, second‐order tensor is analyzed near equilibrium from a point of view that includes many proposed models for liquid crystals, liquid crystal polymers, and fiber suspensions. The steady states of the equation near a uniaxial equilibrium are determined for the fluid in slow, steady, affine flows. Formulae are given for locating the limiting (at zero flow) steady states and determining their linearized stability from a reduced set of constitutive parameters. The various possibilities are graphed so that the dynamical behavior of the relaxation equation in this context may be easily seen.