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:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{(a z<sup>2</sup> + b z + c) d/dz&nbsp;&nbsp;|&nbsp;&nbsp;(a,b,c) ∊ ℂ<sup>3</sup>}

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:

(*)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**V**<sub>P</sub> := {P(z) d/dz &nbsp;|&nbsp; P(z) ∊ C[z]}&nbsp;&nbsp; and &nbsp;&nbsp;**V**<sub>R</sub> := {R(z) d/dz &nbsp;|&nbsp; 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**<sub>P</sub> and/or **V**<sub>R</sub> generate well-studied (infinite-dimensional) Lie groups of transformations of ℂ, then what are these groups?

* In any case, are there standard names for the Lie algebras **V**<sub>P</sub> and **V**<sub>R</sub> ?  

* References to the above matters would also be appreciated.