A benevolent Planner wishes to assign an indivisible private good to n claimants, each valuing the object differently. Individuals have quasi-linear preferences. Therefore, the possibility of transfers is allowed. A second-best efficient mechanism is a strategy-proof and anonymous mechanism that is not Pareto dominated by another strategy-proof and anonymous mechanism. In this context, we identify three conditions that are necessary and, together with Voluntary Participation, sufficient for a mechanism to be second-best efficient. This set includes mechanisms that destroy the good at certain profiles. For domains comprising two individuals we provide an explicit characterization of the family of second-best efficient mechanisms.