wow blast from the past! Sorry but I haven't dealt with this stuff in a long time... anyway the answer by Vit Tucek in the post you linked seems pretty good at a glance

Indeed, suvrit, although that's not exactly the solution I was looking for ;-) Actually that's also a useful observation, since that's where my numerical solutions usually tend to...

Oops 2... $N$ needs to be larger than zero (edited the question)!

Oops.. fixed the series definition (although the boundary condition would make sense too).

These are just formal solutions of first order partial differential equations...

Thanks Robert for yet another great explanation and my apologies to all for mildly flipping out like that and wasting everyone's time with a poorly laid out question... Anyway, the flows and specifically their actions on open sets as Ben McKay describes above are what I'm interested in. By the way Ben McKay (and Robert), what exactly do you mean by "finding invariants of the open sets"? Is it related to what Robert writes about the automorphisms of $B$? I can't see how the automorphisms are related to the problem...

Misha: I'm pretty sure the vector field $V$ I defined sends holomorphic functions to holomorphic functions... although it seems to be composed of holomorphic and antiholomorphic parts. Also I suggest you and everyone else here look up the "How to answer" MO FAQ and especially the part where it says "If, say, a theoretical physicist asks a question on MO and fails to correctly express his holomorphic vector fields and dimensionality of Lie groups, try not to be a dick about it". Ben McKay: Exactly! I guess I should have set up the question more carefully...

Eremenko: forget about $Conf(\mathbb H^2)$ but think of $Diff(S^1)$ instead and the corresponding Lie algebra $Vect(S^1)$ (or maybe it's better to denote it as $diff(S^1)$). Then expand $\xi$ as Fourier series. Misha: a holomorphic vector field isn't such that $\bar \partial \xi = 0$ when $\xi : \mathbb C \to \mathbb C$? OK maybe I got that wrong too. Just call it a "vector field" then. I'm currently too tired to completely edit the question. Close it if you will, I'll leave it up to the community.

Ah... I could map $B \to \mathbb D \to B'$ and $\mathbb D$ is invariant under $PSL(2;\mathbb R)$. Silly me! Anyway, I edited the question.