Abstract A method of reducing; a number of diffraction problems to a system of one-dimensional integro-differential equations is proposed based on the method of discontinuous solutions [1, 2] in the case of steady elastic waves. The defect can be either a spherical crack or a thin rigid spherical inclusion. The method is described in detail for the second case. An effective approximate method of solving the corresponding integro-differential equation in the class of functions with non-integrable singularities is proposed in the case of the diffraction of a torsional wave. A numerical realization of the method is given, namely, graphs of the reactive torsional moment (the inclusion is rigidly fixed) as a function of the oscillation frequency and dimensions of the inclusion are drawn, and the same graphs for the amplitude of the torsional oscillations of the inclusion when it is mobile (not fixed).