The relation between systems and associated bundles
  It is shown that a strong system of vector fields on a fiber bundle in the sense of [Modugno, M. Systems of connections and invariant lagrangians. In: Differential geometric methods in theoretical physics, Proc. 15th Int. Conf., DGM, Clausthal/FRG 1986, 518-534 World Scientific Publishing Co. (1987)] is induced from a principal fiber bundle if and only if each vertical vector field of the system is complete. 
Peter W. Michor