The longitudinal KAM-cocycle of a magnetic flow

Let $M$ be a closed oriented surface of negative Gaussian curvature and let $\mathrm{\Omega }$ be a non-exact 2-form. Let $\lambda$ be a small positive real number. We show that the longitudinal KAM-cocycle of the magnetic flow given by $\la \Omega$isacoboundaryifandonlyiftheGaussiancurvatureisconstantand$\Omega$ is a constant multiple of the area form.