  We study the dynamical system of a forced stratified mixing layer at finite
Reynolds number $Re$, and Prandtl number $Pr=1$. 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, slow-growing, linear instability of the
background profiles at finite $Re$, which complicates the dynamics.
