  We discuss a recently proposed geometric method for constructing a nontrivial
Killing tensor of rank two in a foliated spacetime of codimension one that
lifts trivial Killing tensors from slices to the entire manifold. The existence
of nontrivial Killing tensor is closely related to generalized photon surfaces.
The method is illustrated in some known cases and used to construct the
hitherto unknown Killing tensor for the Nutty dyon in dilaton-axion gravity.
