In the paper, boundary integral equations are derived for the problem of an isotropic homogeneous elastic half-plane with a cavity of an arbitrary configuration. The initially straight-line boundary is assumed to be fixed under a given displacement while the cut surface is subjected to a certain load. A concentrated force is also applied at an inner point of the half-plane. Using the complex variable representation and the method of superposition, the problem is reduced to Fredholm integral equations of the first kind for the boundary of the notch.