  We consider a generalization of the notion of a natural mechanical system to
the case of additional forces of gyroscopic type. Such forces appear, for
example, as a result of global reduction of a natural system with symmetry. We
study symmetries in the systems with gyroscopic forces to find out when these
systems admit a global analogue of a cyclic integral. The results are applied
to the problem of the motion of a rigid body about a fixed point in potential
and gyroscopic forces to find the most general form of such forces admitting
the global area type integral.
