Consider a vector field $X$ in $M=(0,1)^3$ with source at $a=(0,1,1)$ and sink at $b=(1,0,0).$
I'm interested in understanding the collection of vector fields. I mainly want to describe the mappings that map flow lines of $X$ to flow lines of $X$ and describe the isometries of $M.$
The transformation I really wish to write down can be non-rigorously described as a mapping that fixes $a,b$ and "rotates" all other points. Think of water in a cube and everything being rotated about one of the cubes long diagonals.
Does $\text{Iso}(M)$ with $a,b$ form a group or some other structure?
One of my attempts has been to consider flow lines that form a foliation of surfaces in $M$ and try to prove that they are integral surfaces of some partial differential equation.
I've been investigating Lorenz surfaces (Lorenz zonoids) in the spirit of Koshevoy and Mosler and trying to coax the definition into a more geometrical flavor.
Do zonoids arise as solutions to partial differential equations?
I think this may help too.
A zonoid is a surface that is the Hausdorf limit of a sequence of zonotopes.