Integrable sigma models with complex and generalized complex structures

Using the general method presented by Mohammedi \cite{NM} for the integrability of a sigma model on a manifold, we investigate the conditions for having an integrable deformation of the general sigma model on a manifold with a complex structure. On a Lie group, these conditions are satisfied by using the zeros of the Nijenhuis tensor. We then extend this formalism to a manifold, especially a Lie group, with a generalized complex structure. We demonstrate that, for the examples of integrable sigma models with generalized complex structures on the Lie groups ${\mathbf{A}}_{\mathbf{4},\mathbf{8}}$ and ${\mathbf{A}}_{\mathbf{4},\mathbf{10}}$, under special conditions, the perturbed terms of the actions are identical to the WZ terms.