On conjugation of solutions to two integrable problems: Rolling of pointed body in the plane

In this paper, the problem of motion of heavy rigid body in the rough plane is studied. The surface of body is such that the axis of rotation has two “points” where the plane tangent to the surface is not specified. The motion is described by two systems of equations; each of them is true for its side of the phase space. One of them is a system of equations that describes rolling of body in the plane without slippage (the body is tangent to the plane by its convex part); the second, is a system of equations that describes motion of body with the fixed point for the Lagrange case (the body is basing on the plane by the point). Questions of existence of global first integrals and potential of the cited one-dimensional system are studied.