Harmonic perturbations with delay of periodic separated variables differential equations

We study the structure of the set of harmonic solutions to perturbed, nonautonomous, $T$-periodic, separated variables ODEs on manifolds. The perturbing term, supposed to be $T$-periodic in time, is allowed to contain a finite delay. Our main result extends those of cite{FS09} and cite{spaSepVar} but it cannot be simply deduced from them: It emerges from of a combination of the techniques exposed in those two papers.