Solutions to the oligopolistic equilibrium problem have been obtained under a variety of concavity and differentiability assumptions about the supply and inverse demand curves. In this paper we devise a search procedure to find all equilibrium points. Differentiability of the functions involved is not required. We show that, even under struct concavity assumptions, the equilibrium is not necessarily unique. Finally we describe a context of derived demand where nondifferentiability occurs naturally, as well as a numerical comparison with another algorithm.