Andriy Regeta

We study Lie subalgebras L of the Lie algebra Vec^c(A^2_C) of all vector fields  of constant divergence of the affine
2-space A^2_C, and we classify those L which are isomorphic
to the Lie algebra aff_2 of the group Aff_2(C) of affine transformations of A^2_C.
We show that the following statements are equivalent:

(a) The Jacobian Conjecture holds in dimension 2;

(b) All Lie subalgebras L of Vec^c(A^2_C) isomorphic to aff_2 are conjugate under Aut(A^2_C);

(c) All Lie subalgebras L of Vec^c(A^2_C) isomorphic to aff_2 are algebraic.

In addition, we prove the that there exist canonical homomorphisms

Aut(C[x; y]) \to Aut Lie(Vec(A^2_C)) \to Aut Lie(Vec^c(A2_C)) \to Aut Lie(Vec^0(A^2_C))

and they are all isomorphisms.