For a two-dimensional deformation of linear anisotropic elastic materials, the analysis requires computation of certain eigenvalues p that are the roots of a sextic algebraic equation whose coefficients depend only on the elastic constants. It is known that the sextic equation reduces to a cubic equation in p² for materials of monoclinic or higher symmetry with a symmetry plane at xi = 0 or at x2 = 0. The advantage of having a cubic equation in p² is not only that p can be obtained analytically. In many cases, the solution to an anisotropic elasticity problem is much simplified. The purpose of this paper is to present other anisotropic elastic materials for which the sextic equation is a cubic equation in p². These materials may not possess a plane of symmetry. The author shows that as few as two restrictions on the elastic constants are sufficient to deduce these materials.