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In this paper, new representations of three-dimensional asymmetric stress tensor and the corresponding form of the differential equilibrium equations are given. Asymmetric theories of solid mechanics continues to attract attention in connection with the necessity of mathematical modelling of the mechanical behaviour of the advanced materials. The study is restricted to such asymmetric second rank tensors, for which it is still possible to keep the notion of real eigenvalues, but not to accept the mutual orthogonality of the directors of the principal trihedron. The exact algebraic formulation of these asymmetry conditions is discussed. The study extends the dyadic tensor representations of the symmetric stress tensor based on the notion of asymptotic directions. The obtained results are a clear evidence in favor of algebraic hyperbolicity both the symmetric and asymmetric second rank tensors in three-dimensional space.