Computed results are presented from a Reynolds stress, stretched laminar flamelet model, for premixed swirling combustion in a rotating matrix burner. These are in good agreement with flame photography and coherent anti-Stokes Raman spectroscopy (CARS) temperature measurements. The principal variable is the equivalence ratio, phi, while the swirl number and mean axial entry velocity remain constant. At higher equivalence ratios, the flame is stabilized by hot gas in both the inner and outer recirculation zones. As phi is reduced below 0.6 another steady-state solution appears, in which the flame is stabilized only by the hot gas in the inner zone. In this regime, the experiments show unstable combustion with low-frequency oscillations between the two states. This instability is shown to originate in flame quench in the regime between the two recirculations. At the values of phi below 0.6, most of the NO produced originates as prompt NO in the reaction zone. When phi is reduced to 0.56, there is only one steady-state solution. While the investigations were of a laboratory scale burner, some of the general findings are relevant to practical combustors.