I'm reading the following article by Newman

http://scitation.aip.org/content/aip/journal/jmp/4/7/10.1063/1.1704018

about the generalization of the Schwarzschild metric. My question is the following: after the author obtained the tetrad (resp. the metric) on page 5 of the article, he then proceeds to write down the corresponding four killing fields $\xi$ satisfying $\nabla_{\nu}\xi_{\mu}+\nabla_{\mu}\xi_{\nu}=0$. How can one obtain these Killing fields from the tetrad components? That is, I can check explicitly that they really satisfy the Killing equation but I don't know how to get them to start with.

My question is also valid in a general context: given a null tetrad, how does one calculate the Killing fields?

Any hints will be appreciated.