The integral equation of the axisymmetric contact problem, obtained in a new form, is used to investigate the case when thin smooth rigid axisymmetric wedges (posts) are inserted into the bodies making contact along their common axis of symmetry. Singularities are established in the solution, particularly the possibility of the origination of negative drawing together of the bodies and the annular pressure domain under the action of a flat stamp of circular planform on the half-space.