  We show that BPS-impurity theories may support BPS kink-kink solutions i.e.,
an energetically degenerated family of solutions describing two kinks at any
mutual distance. This requires a singular impurity. As an example we consider
the sine-Gordon and $\phi^6$ models coupled with such a BPS impurity.
Interestingly, obtained solutions are identical to double sine-Gordon kinks and
Christ-Lee kinks respectively.
  We also study the spectral flow on the moduli space. All the modes have an
odd number of nodes to cancel the singularity of the impurity.
