By taking values in a commutative subalgebra g l n , C , we construct a new generalized Z n -Heisenberg ferromagnet model in (1+1)-dimensions. The corresponding geometrical equivalence between the generalized Z n -Heisenberg ferromagnet model and Z n -mixed derivative nonlinear Schrödinger equation has been investigated. The Lax pairs associated with the generalized systems have been derived. In addition, we construct the generalized Z n -inhomogeneous Heisenberg ferromagnet model and Z n -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.