This paper re-examines the conditions for the existence of local stationary sunspot equilibria (SSE) in the standard OLG model from a broader perspective than before. We say that local SSE exist around a steady state of a given OLG economy if, in any arbitrarily small neighborhood of the steady state, we can find a SSE of a "nearby" economy. We show that when the domain where "nearby" economies may lie is defined by agents' endowments and probabilities, the indeterminacy of the steady state remains both necessary and sufficient for the existence of local SSE. On the other hand, when the domain of economies is defined by agents' preferences and probabilities, local SSE may exist even around determinate steady states.  We also show that if a slightly weaker notion of distance is used to identify "nearby' economies. SSE in the vicinity of a steady state equilibrium generically.