  We study the Abrikosov-Nielsen-Olesen string in N=2 supersymmetric QED with
N=2-preserving superpotential, in which case the Abrikosov string is found to
be 1/2-BPS saturated. Adding a quadratic small perturbation in the
superpotential breaks N=2 supersymmetry to N=1 supersymmetry. Then the
Abrikosov string is no longer BPS saturated. The difference between the string
tensions for the non-BPS and BPS saturated situation is found to be negative to
the first order of the perturbation parameter.
