We examine the dynamics of a bright solitary wave in the presence of a repulsive or attractive localized “impurity” in Bose–Einstein condensates (BECs). We study the generation and stability of a pair of steady states in the vicinity of the impurity as the impurity strength is varied. These two new steady states, one stable and one unstable, disappear through a saddle-node bifurcation as the strength of the impurity is decreased. Soliton dynamics is also studied, including cases where the soliton is offset from one of the relevant fixed points. The numerical results are corroborated by theoretical calculations which are in very good agreement with the numerical findings.