For the analysis of a two-dimensional steady-state motion such as the surface wave in an anisotropic elastic half-space, the Stroh formalism has always been employed. The solutions are in terms of the elastic stiffnesses C αβ . The Lekhnitskii formalism for elastostatics that provides the solutions in terms of the reduced elastic compliances s αβ is not applicable for two-dimensional steady-state motion. We present a new modified Lekhnitskii formalism in the style of Stroh that can be employed for analyzing two-dimensional steady-state motion. In contrast to the Stroh formalism for which one computes the eigenvector b in terms of the eigenvector a, the new modified Lekhnitskii formalism can compute the eigenvector b without computing the vector a. This feature is attractive in the study of surface waves because the vector b is related to the surface traction. The vanishing of the surface traction at the boundary of the half-space is the key in the surface wave theory. Application to one-component surface waves shows that the conditions for such waves are easily deduced. Motivated by the new modified Lekhnitskii formalism we show that an eigenrelation for the vector b can also be derived for the Stroh formalism.