A class of self-dual and geodesically complete spacetimes with maximally superintegrable geodesic flows is constructed by applying the Eisenhart lift to mechanics in pseudo-Euclidean spacetime of signature (1, 1). It is characterized by the presence of a second rank Killing tensor. Spacetimes of the ultrahyperbolic signature (2, q) with q > 2, which admit a second rank Killing tensor and possess superintegrable geodesic flows, are built.

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It is shown that the near horizon geometry of a generic extreme regular black hole solution of Einstein gravity coupled to nonlinear electrodynamics is described by the AdS 2 × S 2 spacetime.