The frictional contact between two solids, one of which having a periodically grooved surface, under the action of normal and shear load is investigated. The interface between the contacting solids consists of a periodic array of gas-filled gaps and a periodic array of contact regions, where stick-slip contact occurs. The corresponding plane contact problem is reduced to a set of two singular integral equations. A solution to the contact problem is obtained for a certain shape of the grooves. The analysis of dependences of contact parameters of the solids on the applied load and gas pressure is carried out.