Chaplygin ball over a fixed sphere: explicit integration
  We consider a nonholonomic system describing a rolling of a dynamically non-symmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel--Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconical) coordinates on the Poisson sphere, which can be useful in other integrable problems.   Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time. 
A. Borisov, Yu. Fedorov, I. Mamaev