Abstract: Pattern-forming systems admitting a conservation law structure occur naturally in hydrodynamic stability problems with a free surface. An example is the Bénard-Marangoni problem, which models a free-surface fluid on a heated bottom surface. As a parameter increases, the homogeneous ground state of the system destabilizes and spatially periodic patterns bifurcate. These patterns often arise in the wake of an invading heteroclinic front which connects the unstable ground state to the periodic state. This behavior is modeled by a modulating traveling front solution. In this talk, I present an existence result for modulating fronts in a Swift-Hohenberg equation which is coupled to a conservation law. I outline the proof, which is based on spatial dynamics and center manifold reduction, and discuss new challenges, which arise from an additional neutral mode at Fourier wave number k=0.