The general problem of motion of a holonomic conservative mechanical system with cyclic degrees of freedom is considered. A method is introduced to obtain generalizations of such system by adding a number of additional arbitrary parameters in the structure of that system. Although those parameters invoke new physical effects in the generalized system the solution of the latter is always obtained from that of the original system by means of a simple transformation that preserves the Routhian equations of motion. This means, in particular, that from any integrable system with k cyclic degrees of freedom we obtain a k-parameter family of integrable systems, physically generalizing that system.The method is applied to problems of rigid body dynamics, which allow certain types of cyclic variables. Several new integrable problems are obtained as generalizations of known ones. In most cases, the new systems are rare examples of integrable problems of motion involving electrically charged and magnetized rigid bodies under certain combinations of gravitational, magnetic, electric and Lorentz electromagnetic forces.