Adaptation of the Jellett Integral to the Case of Rolling Friction

In his book A Treatise on the Theory of Friction, Jellett considered the problem of tippe top overturn on a rough base. A key to the solution was the first integral of motion, which is invariant to law of friction; the dot product of the kinetic moment and the radius-vector of the point of contact are constant. The possibilities of generalization of this approach to the case in which the friction force is complemented by a moment of rolling friction are considered.