Summary: In the present paper, the issues of the approximate solution of singular integral equation and pair systems of integral equations containing fixed-singularity are studied. The studied integral equations are obtained from the anti-plane problems of the elasticity theory for a composite (piece-wise homogeneous) orthotropic (in particular, isotropic) plane slackened by crack when it reaches or intersects the dividing boundary at the right angle. Algorithms of an approximate solution are designed by the collocation method, namely the method of discrete singularities. In both cases, (when the crack reaches or crosses the dividing border) behaviour of the solutions is studied and the stress intensity coefficients at the ends of the crack are calculated. Results of numerical computations are demonstrated. According to the obtained results, hypothetical predictions of the propagation of crack are made.