The inhomogeneous Suslov problem

We consider the Suslov problem of nonholonomic rigid body motion with inhomogeneous constraints. We show that if the direction along which the Suslov constraint is enforced is perpendicular to a principal axis of inertia of the body, then the reduced equations are integrable and, in the generic case, possess a smooth invariant measure. Interestingly, in this generic case, the first integral that permits integration is transcendental and the density of the invariant measure depends on the angular velocities. We also study the Painlevé property of the solutions. •We consider the Suslov problem of nonholonomic rigid body motion with inhomogeneous constraints.•We study the problem in detail for a particular choice of the parameters that has a clear physical interpretation.•We show that the equations of motion possess an invariant measure whose density depends on the velocity variables.•We show that the reduced system is integrable due to the existence of a transcendental first integral.•We study the Painlevé property of the solutions.