  We investigate the equations of motion in the four-dimensional
non-anticommutative N=2 supersymmetric U(1) gauge field theory, in the search
for BPS configurations. The BPS-like equations, generalizing the abelian
(anti)self-duality conditions, are proposed. We prove full solvability of our
BPS-like equations, as well their consistency with the equations of motion.
Certain restrictions on the allowed scalar field values are also found.
Surviving supersymmetry is briefly discussed too.
