  We find a family of (half) self-dual impurity models such that the self-dual
(BPS) sector is exactly solvable, for any spatial distribution of the impurity,
both in the topologically trivial case and for kink (or antikink)
configurations. This allows us to derive the metric on the corresponding
one-dimensional moduli space in an analytical form. Also the generalized
translational symmetry is found in an exact form. This symmetry provides a
motion on moduli space which transforms one BPS solution into another. Finally,
we analyse exactly how vibrational properties (spectral modes) of the BPS
solutions depend on the actual position on moduli space.
  These results are obtained both for the nontrivial topological sector (kinks
or antikinks) as well as for the topologically trivial sector, where the motion
on moduli space represents a kink-antikink annihilation process.
