  We present the construction of the full set of eigenvectors of the open ASEP
and XXZ models with special constraints on the boundaries. The method combines
both recent constructions of coordinate Bethe Ansatz and the old method of
matrix Ansatz specific to the ASEP. This "matrix coordinate Bethe Ansatz" can
be viewed as a non-commutative coordinate Bethe Ansatz, the non-commutative
part being related to the algebra appearing in
