A homogeneous isotropic perfectly elastic body is subjected to a large static pure homogeneous deformation with two of the principal extension ratios equal. An infinitesimal deformation is superimposed on the large deformation. The conditions for strong ellipticity of the system of equilibrium equations for the infinitesimal deformation are obtained. These conditions are examined within the context of uniqueness or non-uniqueness of the displacement boundary-value problem for the infinitesimal deformation. It is found that the conditions of strong ellipticity are sufficient but not necessary for uniqueness.