New integrable problem of classical mechanics

Complete integrability in Liouville's sense is proven for rotation of an arbitrary rigid body with a fixed point in a Newtonian field with an arbitrary homogeneous quadratic potential. A consequences is the complete integrability of rotation of a rigid body with fixed center of mass in the field of arbitrary sufficiently remote objects (in the second approximation). Explicit formulae are obtained expressing angular velocities of the rigid body in terms of -functions for Riemannian surfaces. Integrable cases are found for rotation of a rigid body in nonlinear Newtonian potential fields.