  We discuss the existence of Killing tensors for certain (physically
motivated) stationary and axially symmetric vacuum space-times. We show
nonexistence of a nontrivial Killing tensor for a Tomimatsu-Sato metric (up to
valence 7), for a C-metric (up to valence 9) and for a Zipoy-Voorhees metric
(up to valence 11). The results are obtained by mathematically completely
rigorous, nontrivial computer algebra computations with a huge number of
equations involved in the problem.
