This paper is concerned with the construction of the fifth-order generalized Heisenberg supermagnetic models. We also investigate the integrable structure and properties of the supersymmetric systems. We establish their gauge equivalent equations with the gauge transformation for two quadratic constraints, i.e., the super fifth-order nonlinear Schrödinger equation and the fermionic fifth-order nonlinear Schrödinger equation, respectively. Substituting (13) and (18) into (31), we findwhere Σ = diag(1, 1, 2).Repeating the process of constraint I, naturally, we obtain J (0) 0

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.

This paper is concerned with the construction of the fifth-order inhomogeneous Heisenberg supermagnetic models. Moreover, the Lax representations of the models are presented. By means of the gauge transformation, we establish their gauge equivalent equations with different quadratic constraints, i.e., the super and fermionic fifth-order inhomogeneous nonlinear Schrödinger equations, respectively. In addition, we investigate their Lax representations and Bäcklund transformations from which the solutions of the super integrable systems have been discussed.