Abstract Analytical solutions for superadiabatic filtration combustion of lean methane-air mixtures in a monolithic porous burner are sought. The one-dimensional, local volume-averaged equations of energy and species conservation that assume a non-thermal equilibrium (i.e., the two-medium treatment), are converted by a coordinate transformation using a combustion wave speed and solved to obtain close-form solutions. A parametric examination varying inlet gas velocity, fuel equivalence ratio, porosity, and thermal conductivity and diffusivity of the solid phase of the porous burner proves the validity of the analytical solutions which are in an excellent agreement with the numerical benchmark. The analytical solutions depict the key features of the filtration combustion such as non-thermal equilibrium between the solid and gas phases, superadiabatic flame temperature, and internal heat recirculation between solid and gas phases of the porous burner.