  The form factor $K(\tau)$ is calculated analytically to the order $\tau^3$ as
well as numerically for a rectangular billiard perturbed by a $\delta$-like
scatterer with an angle independent diffraction constant, $D$. The cases where
the scatterer is at the center and at a typical position in the billiard are
studied. The analytical calculations are performed in the semiclassical
approximation combined with the geometrical theory of diffraction. Non diagonal
contributions are crucial and are therefore taken into account.
  The numerical calculations are performed for a self adjoint extension of a
$\delta$ function potential.
  We calculate the angle dependent diffraction constant for an arbitrary
perturbing potential $U({\bf r})$, that is large in a finite but small region
(compared to the wavelength of the particles that in turn is small compared to
the size of the billiard). The relation to the idealized model of the
$\delta$-like scatterer is formulated. The angle dependent diffraction constant
is used for the analytic calculation of the form factor to the order $\tau^2$.
If the scatterer is at a typical position, the form factor is found to reduce
(in this order) to the one found for angle independent diffraction. If the
scatterer is at the center, the large degeneracy in the lengths of the orbits
involved leads to an additional small contribution to the form factor,
resulting of the angle dependence of the diffraction constant. The robustness
of the results is discussed.
