Abstract: Based on the idea of functionally invariant solutions to the wave equation, invariant displacement solutions to the anti-plane equation of motion for orthotropic bodies are presented. The general representations of the solutions are derived for problems with arbitrary index of self-similarity. The problem under displacement boundary conditions of dissymmetrical extension crack at the interface between orthotropic media is transformed to a single unknown function, which only need satisfy the specific boundary conditions of a given problem. Examples are given to illustrate the present method, in which the solutions for arbitrary complex displacement boundary conditions are obtained based on linear superposition.