  In this paper, we study the inverse scattering problem for energy-dependent
Schr\"{o}dinger equations on the half-line with energy-dependent boundary
conditions at the origin. Under certain positivity and very mild regularity
assumptions, we transform this scattering problem to the one for non-canonical
Dirac systems and show that, in turn, the latter can be placed within the known
scattering theory for ZS-AKNS systems. This allows us to give a complete
description of the corresponding scattering functions S for the class of
problems under consideration and justify an algorithm of reconstructing the
problem from S
