  Some years ago Koutras presented a method of constructing a conformal Killing
tensor from a pair of orthogonal conformal Killing vectors. When the vector
associated with the conformal Killing tensor is a gradient, a Killing tensor
(in general irreducible) can then be constructed. In this paper it is shown
that the severe restriction of orthogonality is unnecessary and thus it is
possible that many more Killing tensors can be constructed in this way. We also
extend, and in one case correct, some results on Killing tensors constructed
from a single conformal Killing vector. Weir's result that, for flat space,
there are 84 independent conformal Killing tensors, all of which are reducible,
is extended to conformally flat spacetimes. In conformally flat spacetimes it
is thus possible to construct all the conformal Killing tensors and in
particular all the Killing tensors (which in general will not be reducible)
from conformal Killing vectors.
