Consider the vector field $w=(x\log x,-y\log y,-z\log z)$ for $0<x,y,z<1.$ I'm wondering if $w$ is Killing for some semi-Riemannian metric.
If we consider a lower dimensional version of $w$ i.e. $v=(x\log x,-y\log y)$ then it's not hard to show that $v$ is Killing and preserves $g=\frac{dxdy}{xy}.$ Furthermore it can be shown that the pair $(M,g)$ is equivalent to $\Bbb M^{1,1}$ (Minkowski plane).
A quick sketch of how to see the equivalence is to solve for the integral curves of $v$ and then inspect them, revealing that they are rectangular hyperbolae under a change in coordinates. Or you can start with the Minkowski metric in standard coordinates, perform a rotation, then pushforward the metric to get $g.$
A similar approach with $w$ yields integral curves of the form $(e^{-t_1e^{s}},e^{-t_2e^{-s}},e^{-t_3e^{-s}})$ parametrized by $s\in (-\infty,\infty)$ and parameters $t_1,t_2,t_3.$
Inspecting this parametrisation leads me to believe that these integral curves foliate a semi-Riemannian manifold that I think would be equivalent to $\Bbb M^{2,1}.$
Am I on the right track to showing that the integral curves of $w$ foliate a semi-Riemannian manifold equivalent to a component of Minkowski $3-$space?