The profiles of conical bodies for which the position of the center of pressure in a supersonic flow with symmetry plane does not depend on the flow parameters are considered. The theoretical investigation of the aerodynamic characteristics of circular cones [1] has shown that their center of pressure does not depend on the angle of attack α when the shock wave is attached to the apex of the cone. It was established experimentally in [2, 3] for star-shaped bodies that the position of the center of pressure for such bodies hardly changes in a wide range of Mach numbers and angles of attack.