Prescribe a map:
$$\Psi:\zeta^2 \to \Bbb T^2$$
which gives a transformation of $\zeta$-space to the "square" flat torus, by identifying $(x,y)\sim (x+1,y)\sim(x,y+1).$
Let $(\zeta^2,g)$ with $g=\frac{dxdy}{xy}$ for $x,y \in (0,1).$ Then we have a Cauchy foliation of $\zeta^2$ defined by:
$$ \mathcal{F_s}=\big\lbrace \log x \log y=s: s>0\big\rbrace $$
and a flow tangent to $\mathcal {F}_s$ which is Killing.
Is it possible to impose conditions s.t. the image of $\mathcal{F}_s$ under $\Psi$ is also Killing?