First order system of differential equations for covariant sigma models

A first order system of differential equations is obtained for a covariant sigma model defined on a Riemannian (or pseudo-Riemannian) manifold of arbitrary dimension n. In the case of compact groups, i.e., SO(n), the first order system coincides with the one yielded by topological arguments. Our considerations hold true also for SO(p,q), p+q=n+1, invariance groups. The scale invariance of the problem is discussed