[Stephani](https://www.researchgate.net/publication/227258228_A_note_on_Killing_tensors) states that in 4 dimensions a spacetime admits a non-reducible Killing-Yano tensor only if the Weyl tensor either is
of Petrov type D or vanishes. Does this imply that the spacetime also cannot admit a Killing tensor for Petrov types other than D?



Connection of the existence of Killing-Yano tensor and Killing tensor
Other Petrov types can admit Killing tensors, i.e. symmetric tensors $K\_{a b} = K\_{(a b)}$ such that $\nabla\_{(a} K\_{b c)} = 0$. A type N vacuum example is given in Example 3 of [A. J. Keane and B. O. J. Tupper, "Killing tensors in pp-wave spacetimes"](https://arxiv.org/abs/1011.6401), which has the metric
\begin{equation}
ds^2 = - 2 du \, dv - (y^2 - z^2) \, du^2 + dy^2 + dz^2 ,
\end{equation}
which admits the irreducible Killing tensor (correcting a factor of 2 at the end of their expression)
\begin{equation}
K\_{a b} \, dx^a \, dx^b = [2 v - u (y^2 - z^2)] \, du^2 - y \, du \, dy - z \, du \, dz - 2 u \, du \, dv + 2 u (dy^2 + dz^2) .
\end{equation}


