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In this paper, we consider a three-node, two-way relay system with digital network coding. The aim is to minimize total energy consumption while ensuring queue stability at all nodes, for a given pair of random packet arrival rates. Specifically, we allow for a set of transmission modes and solve for the optimal fraction of resources allocated to each mode. First, we formulate and solve the static-channel problem, where all link gains are constant over the duration of transmission. Then, we solve the fading-channel problem, where link gains are random. We call the latter the ergodic energy efficiency problem and show that its solution has a water-filling structure. Finally, we provide a detailed analysis of the queues at each node when a random scheduling method that closely approximates the theoretical design is used.