Summary: We consider the long standing problem of integrability of Poisson equations describing spatial motion of a rigid body in the classical nonholonomic Suslov problem. We obtain necessary conditions for their solutions to be meromorphic and show that, under some further restrictions, these conditions are also sufficient. This leads to a family of explicit meromorphic solutions, which correspond to rather special motions of the body in space. We also give explicit extra polynomial integrals in this case. In the more general case (but still under a restriction), the Poisson equations are transformed into a generalized third-order hypergeometric equation. A study of its monodromy group allows us also to calculate the ‘scattering’ angle, i.e. the angle between the axes of limit permanent rotations of the body in space.