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An integral equations method for a three-dimensional crack in a finite or infinite body is achieved by means of Kupradze potentials. Surface and through cracks can be studied according to this approach with only the assumption that the body has a linear, elastic, homogeneous and isotropic behavior. Both singular surface integrals and line integrals appear in the derived equations. For surface and through cracks, the line integral is taken on a part of the crack boundary. The use of our integral equations to the particular problem of an embedded plane crack leads to those formulated by Bui. Another application is devoted to a through crack in a circular cylinder.