Consider the space of diffeomorphisms between $X$ and $Y$:
$$\mathrm{Diff}(X,Y)$$
For $X,Y:=[0,1]^3,$
such that $\mathrm{Diff}(X,Y)$ are maps preserving the smooth structure on the class of all closed embedded surfaces $S\subset [0,1]^3$ satisfying $p\in S$ with $p=(0,1,1)$ and $q\in S$ with $q=(1,0,0)$ and $\mathrm{Diff}(X,Y)$ fixing $p,q.$ Say any candidate class of surfaces forms a smooth foliation of $[0,1]^3$ where the foliation is singular due to a single leaf having dimension $1.$
For a given class $K \subset S$ take the collection of tangent vector fields to each leaf in $K$ whose flow maps from $p$ to $q$ and forms a $3$-vector field. Can anyone give me an example of such a $3$-vector field?