We derive the master function governing the component action of the four-dimensional non-anticommutative (NAC) and fully N=2 supersymmetric gauge field theory with a non-simple gauge group U(2)=SU(2)xU(1). We use a Lorentz singlet NAC-deformation parameter and an N=2 supersymmetric star (Moyal) product which do not break any of the fundamental symmetries of the undeformed N=2 gauge theory. The scalar potential in the NAC-deformed theory is calculated. We also propose the non-abelian BPS-type

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... ations in the case of the NAC-deformed N=2 gauge theory with the SU(2) gauge group, and comment on the SU(3) case too. The NAC-deformed field theories can be thought of as the effective (non-perturbative) N=2 gauge field theories in a certain (scalar only) N=2 supergravity background.