The so-called DNSS points of indifference are of interest because they give decision makers in optimal control problems a choice between following either of two or more trajectories while still achieving optimality. Usually they are described in terms of initial conditions, so that if the system starts at a DNSS point, the decision maker can proceed in either of two or more directions. Here we present a model that has an entire curve of indifference points away from which the decision maker can move in only one direction but does so by choosing either of two trajectories that initially coincide in the state space but later diverge, approaching different long-run steady states.