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Although we have yet to determine whether the DAMA data represents a true discovery of new physics, among such interpretations inelastic dark matter (IDM) can match the energy spectrum of DAMA very well while not contradicting the results of other direct detection searches. In this paper we investigate the general properties that a viable IDM candidate must have and search for simple models that realize these properties in natural ways. We begin by determining the regions of IDM parameter spacedoi:10.1088/1126-6708/2009/05/076 fatcat:gl5zcggpxfcq3g5dovgj6gbsci