  We investigate the perfect transmission resonances (PTRs) of perturbed 1D
finite periodic systems with mirror symmetric cells. The unperturbed scattering
region consists of $N$ identical cells and the related transmission spectrum
possesses at least $N-1$ PTRs in each pass band of the Bloch dispersion of the
unit cell. On the other hand, the perturbation is breaking the periodicity and,
a priori, is able to eliminate all the PTRs. We show how PTRs could still
appear in the perturbed case with a suitable design of the perturbation. We
also reveal a connection between two apparently independent PTRs, a connection
that lies in the symmetry of the finite Kronig-Penney systems and which implies
that if one PTR is preserved then another one, among the $N-1$ PTRs in a pass
band, is necessarily also preserved.
