Question: What are the / some compact subgroups of $Diff(R^n)$, where this is the group of self-diffeomorphisms of n dimensional real Euclidean space? My broader goal is to be able to do some statistics with these groups on my computer, so I want compact subgroups which it is easy to draw from according to their Haar probability measure, and for which the action of individual elements is easy to compute.
I know that if we restrict to linear transformations, the maximal compact subgroup is $O(n)$. This produces many other compact subgroups by conjugation inside of the diffeomorphism group, or by various adhoc tricks for manipulating or combining linear actions (for example, you can diffeomorphically embed $R^n \cup R^n $ inside $R^n$, with complement of the image the y-axis, and then reflect the action of some given compact linear group across the y-axis...).
There are also lots of compact subgroups given by (or perhaps generated) periodic flows of vector fields, but I don't know a good way to describe or compute with these.
However, neither of these ways of producing other subgroups satisfies me.
I want to know if there are other compact subgroups, which can be "easily" described and are mathematically meaningful. In particular, I'm interested in the compact subgroups of the subgroup $Aut_{Reg}(R^n) \subset Diffeo(R^n)$ given by invertible polynomial maps - such as $(x,y) \to (x, y + x^2)$, though I think this element cannot live in a compact subgroup, because the orbit of its action on $(1,0)$ is not precompact. (I'm happy with invertible polynomial maps that only have rational inverses, such as $(x,y) \to (x,y(1+x^2))$, or invertible, everywhere defined rational maps -- I don't care about the complex points in this picture.)
I am interested in the case when $n= 3$, and am especially interested in situations when one can computation draw and apply a random diffeomorphism from such a compact subgroup. For example, for the orthogonal group, it is computationally easy to draw a random operator by means of QR decompositions / Gram-Schmidt,and of course it is easy to compute the result of its action.