Abstract Integral equations are obtained for problems of the elastic equilibrium of a spatial wedge, weakened by a planar strip cut located in the median half-plane of the wedge, for different boundary conditions on the faces of the wedge. A known normal load, which is symmetrical about the plane of the cut is applied to the edges of the cut. In the case when the cut reaches the edge of the wedge, the solution of the problem using the method of paired integral equations reduces to Fredholm integral equations of the second kind with symmetric kernels. The possibility of the cleavage of the wedge along an edge  is shown. A simple formula which is convenient for applications is found for the normal stress intensity factor at one end of the cut. Calculations for various wedge aperture angles are carried out using this formula.