  The statistics of energy levels of a rectangular billiard, that is perturbed
by a strong localized potential, are studied analytically and numerically, when
this perturbation is at the center or at a typical position. Different results
are found for these two types of positions. If the scatterer is at the center,
the symmetry leads to additional contributions, some of them are related to the
angular dependence of the potential. The limit of the $\delta$-like scatterer
is obtained explicitly. The form factor, that is the Fourier transform of the
energy-energy correlation function, is calculated analytically, in the
framework of the semiclassical geometrical theory of diffraction, and
numerically. Contributions of classical orbits that are non diagonal are
calculated and are found to be essential.
