  2D dilaton (super-)gravity contains a special class of solutions with
constant dilaton, a kink-like solution connecting two of them was recently
found in a specific model that corresponds to the KK reduced 3D Chern-Simons
term. Here we develop the systematics of such solutions in generalized 2D
dilaton gravity and supergravity. The existence and characteristics thereof
essentially reduce to the discussion of the conformally invariant potential W,
restrictions in supergravity come from the relation W=- 2 w^2. It is shown that
all stable kink solutions allow a supersymmetric extension and are BPS therein.
Some examples of polynomial potentials are presented.
