Although a national live-donor kidney exchange program is being launched in the US, the kidney shortage is increasing faster than ever. A new solution paradigm is able to incorporate <i>compatible pairs</i> in exchange. In this paper, we consider an exchange framework that has both compatible and incompatible pairs, and patients are indifferent over compatible pairs. Only two-way exchanges are permitted due to institutional constraints. We explore the structure of Pareto-efficient matchings in this framework. The mathematical structure of this model turns out to be quite novel. We show that under Pareto-efficient matchings, the same number of patients receive transplants, and it is possible to construct Pareto-efficient matchings that match the same incompatible pairs while matching the least number of compatible pairs. We non-trivially extend the famous Gallai-Edmonds Decomposition in the combinatorial optimization literature to our new framework.