Summary: The paper examines, within the framework of a multi-dimensional one-step forward-looking model, a special category of rational expectations equilibria. Their support is infinite with two accumulation points (steady states); the stochastic motion of the system is of random-walk type. A general strategy for an existence proof -- associated with the study of a dynamical system -- stresses necessary conditions. In a simple overlapping-generations model, the proof is made complete -- no backwards bending labour supply is required in the purse sunspot case. By continuity, heteroclinic random walk equilibria are also shown to exist when shocks are real.