  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.
