On the bi-Hamiltonian structure of the Chaplygin and Borisov–Mamaev–Fedorov systems at a zero constant of areas. II

The main aim of the second part of the paper is a construction of the rational potentials, which may be added to the Hamiltonians of the Chaplygin and Borisov–Mamaev–Fedorov systems without loss of integrability. All these potentials may be considered as natural nonholonomic generalizations of the standard separable potentials associated with an elliptic (or sphero-conical) coordinate system on the sphere.