In studying the transformation groups generated by holomorphic vector fields V(z) d/dz on ℂ, I've noticed the (surely well-known) fact that the complex quadratic vector fields:

{(a z^{2} + b z + c) d/dz | (a,b,c) ∊ ℂ^{3}}

form precisely the Lie algebra whose nonzero elements generate the linear fractional transformations, i.e., PSL(2,ℂ).

- Is there some underlying reason for this? (Beyond direct calculation, which provides an easy proof.)

Other than the further Lie algebras {0}, {a d/dz}, and {(a z + b) d/dz} over ℂ of trivial, constant, and linear (affine) vector fields, there seem to be no other polynomial finite-dimensional Lie algebras of vector fields on ℂ.

- Is there some easy-to-explain reason for this?

The next "simplest" such polynomial Lie algebras of vector fields on C *seem* to be those defined by all polynomial and all rational functions:

(*) **V**_{P} := {P(z) d/dz | P(z) ∊ C[z]} and **V**_{R} := {R(z) d/dz | R(z) ∊ C(z)}.

Contrariwise, do there exist finite-dimensional Lie algebras of vector fields on ℂ defined by rational functions -- other than the polynomial ones mentioned above?

In case

**V**_{P}and/or**V**_{R}generate well-studied (infinite-dimensional) Lie "groups" of transformations from open sets of ℂ into open sets, then what are these groups? Properly, these are**pseudogroups**, but perhaps they behave like Lie groups.

[Note: It's not hard to compute formulas for such transformations -- the flows -- directly from an expression for the vector field in terms of its zeroes (and poles, if any).]

In any case, are there standard names for the Lie algebras

**V**_{P}and**V**_{R}?References to the above matters would also be appreciated.