Because sunspot equilibria seem to be of central importance for an understanding of rational expectations, we seek here to characterize completely a limited class of sunspot equilibria (stationary ones with two possible natural events) in the simplest overlapping generations model of production. We present a sufficient condition for the existence of stationary sunspot equilibria, examine how these are related to strictly periodic equilibria of the same order, and investigate how deterministic stationary equilibria bifurcate to stationary sunspot equilibria. A concluding section examines how our results survive in more general settings.