Consider a transformation $(x, t)\mapsto (x',t') = \left(x^{k}, t^{\frac{1}{k}}\right)$ with $k=e^{\Delta b}.$ The metric it preserves is just that points stay on certain curves. These curves are given by: $$ t=\exp\left(\frac{n}{\log x}\right) $$
So $$n=\log(x)\log(t)$$ is a preserved quantity of the transformation.
Consider three more preserved quantities of similar transformations: $$n=\log(1-x)\log(t) $$
$$ n=\log(x)\log(1-t) $$
$$ n=\log(1-x)\log(1-t) $$
How does one express the combination of all four transformations?
I think it has to do with composition but I can't see how to fit everything together in a way that makes sense. I am trying to describe the combination of the four transformations happening simultaneously so I don't see how composition will work.