  Within the first order formalism static solutions of generic dilaton gravity
in 2D with self-interacting (scalar) matter can be discussed with ease. The
question of (non)existence of Killing horizons is addressed and the interplay
with asymptotic conditions is investigated. Naturally, such an analysis has to
be a global one. A central element in the discussion is the rank of the Jacobi
matrix of the underlying dynamical system. With some (pathological) exceptions
Killing horizons exist only if it equals to 3. For certain self-interactions
asymptotically flat black holes with scalar hair do exist. An example relevant
to general relativity is provided. Finally, generalizations are addressed
including 2D type 0A string theory as a particular example. Additionally, in a
pedagogical appendix the mass definition in dilaton gravity is briefly reviewed
and a unique prescription to fix scaling and shift ambiguity is presented.
