  We present a purely geometric method for constructing a rank two Killing
tensor in a spacetime with a codimension one foliation that lifts the trivial
Killing tensors from slices to the entire manifold. The resulting Killing
tensor can be nontrivial. A deep connection is found between the existence of
such a Killing tensor and the presence of generalized photon surfaces in
spacetime with two Killing vector fields. This technique generates Killing
tensors in a purely algebraic way, without solving differential equations. The
use of our method is demonstrated for Kerr, Kerr-Newman-NUT-AdS metrics and
Kerr-NUT-AdS multicharge gauged supergravity solution.
