  The rolling of a dynamically balanced ball on a horizontal rough table
without slipping was described by Chaplygin using Abel quadratures. We discuss
integrable discretizations and deformations of this nonholonomic system using
the same Abel quadratures. As a by-product one gets new geodesic flow on the
unit two-dimensional sphere whose additional integrals of motion are
polynomials in the momenta of fourth order.
