A study of saturated magnetic island equilibria on the basis of the resistive magneto-hydro-dynamic model is presented. A bifurcation in the sequence of equilibria is found as the ratio of the width of the current layer in the initial (non-reconnected) configuration over the island periodicity length reaches a critical threshold. Below this threshold, spontaneous generation of zonal flows occurs. This result is suggestive of a possible evolution of current sheets in magnetically confined plasmas and may be relevant to the understanding of the suppression of drift-wave turbulence and the formation of internal transport barriers in tokamak experiments.