We study the dynamical system of a forced stratified mixing layer at finite Reynolds number Re, and Prandtl number P r = 1. We consider a hyperbolic tangent background velocity profile in the two cases of hyperbolic tangent and uniform background buoyancy stratifications. The system is forced in such a way that these background profiles are a steady solution of the governing equations. As is well-known, if the minimum gradient Richardson number of the flow, Ri m , is less than a certain critical value Ri c , the flow is linearly unstable to Kelvin-Helmholtz instability in both cases. Using Newton-Krylov iteration, we find steady, two-dimensional, finite amplitude elliptical vortex structures, i.e. 'Kelvin-Helmholtz billows', existing above Ri c . Bifurcation diagrams are produced using branch continuation, and we explore how these diagrams change with varying Re. In particular, when Re is sufficiently high we find that finite amplitude Kelvin-Helmholtz billows exist at Ri m > 1/4, where the flow is linearly stable by the Miles-Howard theorem. For the uniform background stratification, we give a simple explanation of the dynamical system, showing the dynamics can be understood on a two-dimensional manifold embedded in state space, and demonstrate the cases in which the system is bistable. In the case of a hyperbolic tangent stratification, we also describe a new, slowgrowing, linear instability of the background profiles at finite Re, which complicates the dynamics. † Email address for correspondence: jpp39@cam.ac.uk