Summary: In the present article an anti-plane problem of the elasticity theory for a composite (piece-wise homogeneous) orthotropic body weakened by cracks intersecting the interface or reaching it in a right angle is studied. The studied problem is reduced to the singular integral equation (when crack reaches the interface) and system (pair) of singular integral equations (when crack intersects the interface) containing an immovable singularity with respect to the unknown characteristic function of the crack disclosure. Behavior of solutions in the neighborhood of the crack endpoints is studied by the method of discrete singularity with uniform division of an interval by knots. In both cases (crack intersects or reaches the interface) the question of behavior of approximate solutions are investigated. The corresponding algorithms are composed and realized. The results of numerical investigations are presented.