  We reveal the origin and structure of self-consistent source extensions of
integrable equations from the perspective of binary Darboux transformations.
They arise via a deformation of the potential that is central in this method.
As examples, we obtain in particular matrix versions of self-consistent source
extensions of the sine-Gordon, nonlinear Schrodinger, KdV, Boussinesq, KP,
Davey-Stewartson, two-dimensional Toda lattice and discrete KP systems. We also
recover a (2+1)-dimensional version of the Yajima-Oikawa system from a
deformation of the pKP hierarchy. By construction, these systems are
accompanied by a hetero binary Darboux transformation, which generates
solutions of such a system from a solution of the source-free system and
additionally solutions of an associated linear system and its adjoint. The
essence of all this is encoded in universal equations in the framework of
bidifferential calculus.
