The extended Zn-Heisenberg ferromagnet model

By virtue of taking values in a commutative subalgebra [Formula: see text] of Lie algebra [Formula: see text], we construct the [Formula: see text]-Heisenberg ferromagnet model which contains many Heisenberg ferromagnet-type equations. Moreover, we investigate the integrable properties of the [Formula: see text]-Heisenberg ferromagnet model. In terms of the gauge transformation, the gauge equivalent counterpart of the [Formula: see text]-Heisenberg ferromagnet model has been presented. Based on the differential geometry of curves and surfaces, the corresponding geometrical equivalence between the [Formula: see text]-Heisenberg ferromagnet model and [Formula: see text]-nonlinear Schrödinger equation has also been established. Furthermore, we also discuss the [Formula: see text]-generalized inhomogeneous Heisenberg ferromagnet model.