It's simple to construct an analytic codimension one foliation of $\Bbb R^2_{\gt 0}$ with one class of functions. I wonder if there are pros and or cons of using more distinct classes of functions for the foliation.
For example is it possible to have a codimension one foliation of $\Bbb R^2_{\gt 0}$ with distinct analytic classes of functions $f_t(x)\ne g_t(x)\ne h_t(x)$ where $t$ indexes the leaves?
I have found an example of such. However, I still would like to know:
Is there any motivation for building the foliation by maximizing the distinct classes of functions as opposed to constructing a foliation using only one class of functions?
Just thinking about some possible motivations for this, I thought that using one class of functions for the foliation might make for a vector field tangent to the leaves easier to write down and describe. And that's definitely useful.
However, I think introducing variation in the leaves of the foliation could create richer dynamics with the vector field that is tangent to the leaves.
A question that comes to my mind is: Do the collection of some given distinct classes of leaves preserve some global metric?
An example for one class of functions I can provide is $(M,g,\mathcal F)$ with metric tensor $g=dudv$ and foliation comprised of a single class of functions, $\mathcal F=\lbrace t/x:t\in \Bbb R_{\gt 0} \rbrace.$ It can be shown that the infintesimally generated flow tangent to $\mathcal F$ preserves $g$.
However at this moment it's unclear to me how to stitch together mutually distinct one parameter classes of functions that preserve some global metric.