A class of initial-boundary value problems with a conserved first integral is studied. The class of problems includes particular cases of interest. In one parameter limit, the equations arise as a similarity reduction of the Navier-Stokes equations, which has been recently studied for its blow-up properties, and in another, a class of scalar reaction-diffusion equations, modified by the addition of a non-local non-linearity to ensure conservation of the first integral. Once the problem has been stated, it is shown how it may be derived from the Navier-Stokes equations in one parameter limit. Steady-state solutions are then constructed using a rigorous iterative method which we call Crandall Iteration. The steady solution set includes, in a particular parameter limit, those of the Cahn-Hilliard equation. Amplitude expansions and centre manifold theory are employed to analyze the heteroclinic orbits connecting these steady solutions. It is proved that Hopf bifurcation of periodic solutions cannot occur