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We generalize our recent method for constructing Killing tensors of thesecond rank to conformal Killing tensors. The method is intended for foliatedspacetimes of arbitrary dimension $m$, which have a set of conformal Killingvectors. It applies to foliations of a more general structure than in previousliterature. The basic idea is to start with reducible Killing tensors in slicesconstructed from a set of conformal Killing vectors and the induced metric, andthen lift them to the whole manifold. Integrability conditions are derived thatensure this, and a constructive lifting procedure is presented. The resultingconformal Killing tensor may be irreducible. It is shown that subdomains offoliation slices suitable for the method are fundamental photon surfaces ifsome additional photon region inequality is satisfied. Thus our procedure alsoopens the way to obtain a simple general analytical expression for the boundaryof the gravitational shadow. We apply this technique to electrovacuum, and${\cal N}=2,\,4,\,8$ supergravity black holes, providing a new easy way toestablish the existence of exact and conformal Killing tensors.

Slice-reducible conformal Killing tensors, photon surfaces and shadows