Constructing Killing tensors from Killing vectors

Background:

After reading about Carter constant and symmetries in GR, I became interested in Killing tensors.

I tried reading this paper by Alan Barnes, Brian Edgar and Raffaele Rani, discussing conformal Killing tensors. I have some trouble understanding the crux of the paper.

Question:

• Is there a general way to construct Killing tensors, if the Killing vectors are known?

• How would I do this?

• Are there any Killing tensors that can not be constructed from Killing vectors?

Initial guess/motivation for the question:

Initially, I thought Killing tensors could just be formed via $K_{\mu \nu}=k_\mu k'_\nu$, where $K_{\mu \nu}$=Killing tensor, $k_\mu,k'_\nu$=Killing vectors. After reading the above paper, I am no longer sure. The paper discusses conformal Killing tensors and vectors, which may be the source of my confusion.

• I'm now re-reading the article in hopes of learning a bit more. – Otto Oct 29 '15 at 6:59
• This is a short answer But basically Killing tensors can be constructed at least via: $K_{\mu \nu}=k_\mu k_\nu$ This satisfies the Killing tensor equation: $K_{(\mu \nu;c)}=0$ Just by using the Killing vector equation: $k_{(\mu;c)}=0$ – Otto Oct 29 '15 at 7:19