Is this a Killing vector field?

[ergospherical]

There is a gravitational wave spacetime described by$$g = a(u) (x^2-y^2)du^2 + 2du dw + dx^2 + dy^2$$There is one obvious Killing vector field, $\partial/\partial w \equiv \partial_w$. To find some more, it's suggested to try:$$X = xf(u) \frac{\partial}{\partial w} + p(u) \frac{\partial}{\partial x}$$In principle, you could compute the connection coefficients, write out Killing's equation, and verify each component. But that is generally tedious, so instead I thought it may be easier to use directly the condition $L_X g = 0$. Explicitly,\begin{align*}
0 = L_X g &= L_{xf \partial_w} g + L_{p \partial_x} g \\
&= \left[ xf L_{\partial_w} g + d(xf) \wedge i_{\partial_w} g \right] + \left[ p L_{\partial_x} g + dp \wedge i_{\partial_x} g\right] \\
&= d(xf) \wedge i_{\partial_w} g + p L_{\partial_x} g + dp \wedge i_{\partial_x} g \\ \\
&= (fdx + xf'(u) du) \wedge i_{\partial_w} g + p L_{\partial_x} g + p'(u) du \wedge i_{\partial_x} g
\end{align*}
where I used $L_{\partial_w} g = 0$. Here $i_{\partial_w} g$ and $i_{\partial_x} g$ are one-forms with components \begin{align*}
(i_{\partial_w} g)_{\alpha} &= g_{w\alpha} = g_{wu} \delta^u_{\alpha} \\
(i_{\partial_x} g)_{\alpha} &= g_{x\alpha} = g_{xx} \delta^x_{\alpha}
\end{align*}Any ideas how to find conditions on $f(u)$ and $p(u)$ for $X$ to be Killing? Maybe just a case of expanding out the terms, which is probably a good idea to try...

 

[ergospherical]

In fact, you can re-write it as\begin{align*}
0 = L_X g &= (f(u)g_{uu} - p'(u) g_{xx}) dx \wedge du + pL_{\partial_x} g \\
&= \left[ f(u)a(u)(x^2-y^2) - p'(u) \right] dx \wedge du + p L_{\partial_x} g
\end{align*}So you need to arrange$$L_{\partial_x} g = - \frac{1}{p(u)} \left[ f(u)a(u)(x^2-y^2) - p'(u) \right] dx \wedge du$$...? That can't be it...