Asymptotically $AdS_2\times S^2$ metrics satisfying the Null Energy Condition

We find the most general metric ansatz compatible with the results of Galloway and Graf \cite{GG} constraining asymptotically $AdS_2\times S^2$ space-times (and a differentiability assumption), and then study its curvature subject to a variety of geometrical and physical restrictions. In particular we find explicit examples which are asymptotically $AdS_2\times S^2$ metrics, in the sense of \cite{GG}, and which satisfy the Null Energy Condition but which differ from $AdS_2\times S^2$.