Consider a bounded (by a unit cube) real fibered $3$-manifold $M=(0,1)^3.$
Slicing $M$, with planes orthogonal to the faces of the cube, for example, $z=c$ for some positive constant $c\in(0,1)$ yield $\Bbb R^2_{\gt0}$ equipped with flat metric $ds^2=\frac{dx^2}{x^2}+\frac{dy^2}{y^2},$ for $x,y\in (0,1).$ This is diffeomorphic to a component of $\Bbb R^{2}$ (the third quadrant).
This means slicing $M$ by planes orthogonal to the cubes faces results in $\Bbb R_{\gt 0}^2$ with metric $ds^2=\frac{dx^2}{x^2}+\frac{dy^2}{y^2}.$
$M$ must have cusp singularities at $(0,1,1)$ and $(1,0,0).$ This is where the fibers reach singularity simultaneously.
Notice that these flat $2D$ slices admit fibrations - an example being $(\log x)(\log y)=\log^2 s$ where $s$ indexes. Call this specific fibration $E.$
$M$ is a real fibered $3$-manifold. What I'd like to construct is a specific fibration of $M$ which is compatible with $E.$ By compatible I mean that all the slices of $M$ are of the form $E.$
How is this done?