This paper analyzes traveling wave profiles possessing an internal resonance point for a class of systems of partial differential equations describing oxidation and vaporization of liquid fuel in a porous medium when an oxidizer (air) is injected. It is shown that the resonance is characterized by a saddle point of an associated vector field defined on a folded surface in state space. We prove existence and uniqueness of this singularity for an open set of parameters. The singularity yields an extra restriction on wave parameters. This restriction is found explicitly in the physically relevant case of a small ratio of reaction/vaporization rates. We find the general large time asymptotic solutions of the problem as a sequence of waves and show that resonance waves play a key role in determining these solutions. A numerical example is presented.