  We consider an ensemble of cavity-coupled two-level emitters interacting via
full (coherent and dissipative) dipole-dipole interactions. Using an adiabatic
elimination procedure we derive effective equations of motion for a subsystem
consisting of the cavity and a single emitter. Those equations can be used for
schemes to enhance the cavity-coupling of single emitters as shown in [S.
Sch\"{u}tz, J. Schachenmayer, D. Hagenm\"{u}ller, G. K. Brennen, T. Volz, V.
Sandoghdar, T. W. Ebbesen, C. Genes, and G. Pupillo, e-print arXiv:1904.08888].
We analyze limitations of effective subsystem parameters, and study how joint
dissipative decay processes in the subsystem affect cavity-coupling properties
of the single emitter and cavity transmission spectra.
