Coupled systems, each one admitting a family of nondegenerate periodic
solutions, are considered. The period of oscillations in the family is supposed to
depend on a unique parameter. Conditions imposed on weak couplings such that
the coupled system admits a family of periodic solutions, which is similar to that of
subsystems, are found. Differential equations of general form, as well as reversible
mechanical systems, are investigated. The existence of resonant orbits in the N-planet problem with one planet in a quasi-circular orbit is proved.