  We consider the the n-dimensional generalisation of the nonholonomic Veselova
problem. We derive the reduced equations of motion in terms of the mass tensor
of the body and determine some general properties of the dynamics. In
particular we give a closed formula for the invariant measure, we indicate the
existence of steady rotation solutions, and obtain some results on their
stability.
  We then focus our attention on bodies whose mass tensor has a specific type
of symmetry. We show that the phase space is foliated by invariant tori that
carry quasi-periodic dynamics in the natural time variable. Our results enlarge
the known cases of integrability of the multi-dimensional Veselova top.
Moreover, they show that in some previously known instances of integrability,
the flow is quasi-periodic without the need of a time reparametrisation.
