Consider a Lorentzian manifold $(\zeta,g)$ with metric: $$g=\frac{dudv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2}.$$
For $u,v,r,w \in(0,1)$.
It can be derived that:
$$\Omega_s(x,y,z)=\varphi_s(x)\varphi_s(y)\varphi_s(z)\space\space\space\space{s>0}$$
defines a Cauchy foliation of $\zeta$ for $\Omega_s(x,y,z)=e^{\frac{s}{\log x}+\frac{s}{\log y}+\frac{s}{\log z}}$ and that $\Omega$ preserves the metric.
If we do surgery with $\Omega$ and embed it within the Lorentzian manifold with metric $h=dudv-dr^2-dw^2$ then we have $\Omega$ acting as a foliation of a subset of this manifold that longer preserves that metric.
We can write down a linear, bijective, continuous and smooth map between the foliations:
$$T: \Omega \to \chi$$
where $T$ is given by:
$$\int_{(0,1)^3} x^{a}y^{b}z^{c}\Omega_s(x,z,y)~\frac{dxdydz}{xyz}=\chi_s(a,b,c).$$
My objective is to determine a Lorentzian metric which is preserved by $\chi_s(a,b,c).$
In other words, can we find a Killing vector field whose flow maps $\chi$ to itself? And what would be an example of one?
I noticed that $T$ is not only a map between foliations, it defines a symmetric nondegenerate smooth mapping so that we can obtain a semi-Riemannian metric from it.
Using that knowledge I thought to test whether that metric is preserved by the foliation $\chi$, where both the metric and foliation that (possibly) preserves it, are being obtained directly through $T$, but I'm not sure of the correct steps in testing this.