Before revision: We establish several structural results for the set of competitive equilibria in bilateral trading networks with transfers in the case of imperfectly transferable utility and frictions: The lattice theorem, the rural hospitals theorem, the existence of side-optimal equilibria, compactness of the set of equilibria and a group-incentive-compatibility results hold without the assumption of quasi-linear utility in  transfers.
After revision: We establish several structural results for the set of competitive equilibria in bilateral trading networks with transfers in the case of imperfectly transferable utility and frictions: The lattice theorem, the rural hospitals theorem, the existence of side-optimal equilibria, compactness of the set of equilibria and a group-incentive-compatibility results hold without the assumption of quasi-linear utility in transfers. While our results are developed in a trading network model, they also imply analogous (and new) results for exchange economies with combinatorial demand and for two-sided matching markets with transfers.