In this paper we provide a general derivation of the non-thermal (as well as of the thermal) SZ effect in galaxy clusters which is exact in the Thomson limit to any approximation order in the optical depth. Such a general derivation is obtained using the full relativistic formalism and overcoming the limitations of the Kompaneets and of the single scattering approximations. To have a consistent description of the SZ effect in clusters containing non-thermal phenomena, we also evaluate in a consistent way - for the first time - the total SZ effect produced by a combination of thermal and non-thermal electron populations residing in the same environment, like is the case in radio-halo clusters. We discuss both the spectral and the spatial features of the total SZ effect and we provide specific predictions for a well studied radio-halo cluster like A2163. We also discuss the overall SZ effect produced by a combination of different thermal populations residing in the cluster atmosphere. In this context, we discuss how the combined observations of the thermal and non-thermal SZ effect provide constraints on the spectrum of the relativistic electron population and, in turn, on the presence and on the origin of non-thermal phenomena (radio-halo, hard X-ray, EUV excesses) occurring in galaxy clusters
