  It was shown that free energy density of the local flexomagnetoelectric
effect is determined by the four phenomenological constants in case of the
cubic (hexoctahedral) crystal. The well-known single-constant Lifshitz
invariant term is correct only when fixed electric polarization induces the
inhomogeneity of the magnetization. Proposed phenomenological theory was
applied to the magnetic domain walls. The domain wall structure has been
investigated in details. The four-constant phenomenological theory conforms to
the symmetry based predictions (V.G. Bar'yakhtar et al., 1984). The proposed
experimental verification of the four-constant flexomagnetoelectric
phenomenology is a detection of the shift of the N\'eel domain walls under the
strong homogeneous electric field.
