  We consider one-dimensional systems in the presence of a quasi-periodic
perturbation, in the analytical setting, and study the problem of existence of
quasi-periodic solutions which are resonant with the frequency vector of the
perturbation. We assume that the unperturbed system is locally integrable and
anisochronous, and that the frequency vector of the perturbation satisfies the
Bryuno condition. Existence of resonant solutions is related to the zeroes of a
suitable function, called the Melnikov function - by analogy with the periodic
case. We show that, if the Melnikov function has a zero of odd order and under
some further condition on the sign of the perturbation parameter, then there
exists at least one resonant solution which continues an unperturbed solution.
If the Melnikov function is identically zero then one can push perturbation
theory up to the order where a counterpart of Melnikov function appears and
does not vanish identically: if such a function has a zero of odd order and a
suitable positiveness condition is met, again the same persistence result is
obtained. If the system is Hamiltonian, then the procedure can be indefinitely
iterated and no positiveness condition must be required: as a byproduct, the
result follows that at least one resonant quasi-periodic solution always exists
with no assumption on the perturbation. Such a solution can be interpreted as a
(parabolic) lower-dimensional torus.
