Integrability from an Abelian subgroup of the diffeomorphisms group

It has been known for some time that for a large class of nonlinear field theories in Minkowski space with two-dimensional target space the complex eikonal equation defines integrable submodels with infinitely many conservation laws. These conservation laws are related to the area-preserving diffeomorphisms on target space. Here we demonstrate that for all these theories there exists, in fact, a weaker integrability condition which again defines submodels with infinitely many conservation laws. These conservation laws will be related to an Abelian subgroup of the group of area-preserving diffeomorphisms. As this weaker integrability condition is much easier to fulfill, it should be useful in the study of those nonlinear field theories