We demonstrate the existence of both one and two stationary reaction zone structures which arise in filtration combustion in a moving porous medium. This configuration may be important for SHS reactors. Using the narrow reaction zone approximation, we derive approximate analytical expressions for the principal combustion characteristics, including the combustion temperature, the temperature and depth of solid conversion at the first reaction zone, the locations of the reaction zone(s), the inlet and outlet oxidant fluxes, as well as profiles for the spatial distributions of pressure, temperature, and depth of conversion, corresponding to the stationary reaction zone structures. We also determine the region of existence in parameter space for each solution structure. In addition, we numerically simulate the full time-dependent problem, and describe the evolution to steady-state behavior corresponding to the stationary two zone structure, thus demonstrating that the structure is stable. We find rather good agreement between the analytical results and those obtained from the simulations. Finally, various properties of both the one and two zone structures are described.