Helmholtz resonator has been used in industry for a long time to reduce the noise from exhaust system in vehicle or machinery. Numerous investigations have been done in the past to study the effect of a Helmholtz resonator connected to a pipeline. A general procedure for the analysis of curved or flat, thin two dimensional gas cavities such as thin compressor or engine manifolds or so-called thin shell type muffler elements, which can efficiently utilize the limited space of hermetically sealed compressors or small engine compartments, has been developed by the authors, as long as the thickness of the cavities is substantially small compared to the shortest wavelength of interest. However, to the authors' knowledge, a Helmholtz resonator attached to a rectangular thin muffler element, which is similar to a refrigeration compressor muffler, has not been analyzed. In this study, a point impedance method is proposed to solve for the natural frequencies and mode shapes of this combined system. Point impedances are formulated, in analogy to point receptances in structural analysis, from the pressure responses to the volume flow rate inputs at the joint of both Helmholtz resonator and thin muffler element. The frequency characteristic equation is derived and then solved for the natural frequencies and mode shapes. The acoustic four poles are then obtained by substituting the natural frequencies and mode shapes into the general formulations developed by the authors. The results from the analytical method are also compared to the numerical prediction from a three dimensional finite element method.