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By taking values in a commutative subalgebra g푙(푛, ℂ), we construct a new generalized -Heisenberg ferromagnet model in (1+1)-dimensions. e corresponding geometrical equivalence between the generalized -Heisenberg ferromagnet model and -mixed derivative nonlinear Schrödinger equation has been investigated. e Lax pairs associated with the generalized systems have been derived. In addition, we construct the generalized -inhomogeneous Heisenberg ferromagnet model and -Ishimori equation in (2+1)-dimensions. We also discuss the integrable properties of the multi-component systems. Meanwhile, the generalized Z n -nonlinear Schrödinger equation, Z n -Davey-Stewartson equation and their Lax representation have been well studied.

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