Surfaces of M_k^2times mathbb R invariant under a one-parameter group of isometries

We assume that an immersed constant mean curvature surface ${varSigma } looparrowright {M_k} times mathbb R $ satisfies a relation involving the mean curvature, the Gaussian curvature and the angle that the unit vector of the factor $mathbb R $ makes with the normal to the surface. This relation, although given initially in its pointwise form, can be shown to be equivalent to an integral relation. From the assumed relation, it follows that $varSigma $ is invariant under a one-parameter group of isometries of $M_k^2times mathbb{R }$ which are induced by the isometries of $M_k^2$  . An application is made to describe qualitatively those surfaces for which the Abresch-Rosenberg complex quadratic form vanishes.