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to zero. The BPS equations form a system of matrix-valued partial differential equations in terms of the bosonic fields of the theory. One systematic approach to solve them, assuming no fermionic backgrounds, begins by forming various Killing spinor bilin- ears [66, 67]. The BPS equations may then be expressed as a set of coupled first-order equations for these tensor fields, which describe the bosonic background of the solu- tion. This approach was used in [39, 21] to solve the off-shell problem in the AdS2 × S2 (and S3) background. The general solutions to the resulting equations are, however, typically difficult to obtain, and we do not solve this problem of general classification in this paper. Instead, we leverage what is already known about the localization so- lutions in 4d supergravity around the Euclidean AdS2×S2 background [14, 39, 64], by lifting them to five dimensions. This involves the ▇▇▇▇▇▇-▇▇▇▇▇ (KK) lift of AdS2 ×S to AdS3× S2, which we describe in Section 6.1. Note, however, that while the 4d localization manifold has been determined completely, there may be additional solu- tions in 5d that do depend on the KK direction, and that will therefore not emerge from the lift. We postpone the discussion of such solutions to future work. To lift the 4d localization solutions, we use the idea of the 4d/5d off-shell con- nection of [37]. However, as mentioned in the introduction, implementing this idea is not straightforward for the following reasons. Firstly, while the formalism in [37] was developed for Lorentzian supergravities, our 4d/5d connection needs to be adapted to accommodate the Euclidean supergravities in both four and five dimensions. A sub- tlety here, as we will shortly see, is that the 4d Euclidean theory has a redundancy in the choice of reality conditions and correspondingly a redundancy of AdS2×S2 backgrounds, which has no counterpart in the 5d theory. Secondly, recall that the 4d/5d lift produces a five-dimensional background in the ▇▇▇▇▇▇-▇▇▇▇▇ ansatz and so, in order to reach the five-dimensional theory on the supersymmetric twisted torus H3/Z×S2 from the four-dimensional theory on AdS2×S2, we require a mapping of the twisted torus (5.8) into the ▇▇▇▇▇▇-▇▇▇▇▇ frame of AdS3×S2. In Section 6.1 we present the mapping from the ▇▇▇▇▇▇-▇▇▇▇▇ frame to the cylinder frame. The twisted frame can then easily be mapped to the cylinder frame (5.3) by the local coordinate transformation (5.6). In Section 6.2 we review the 4d Euclidean supergravit...