Lie Subalgebras of Vector Fields on Affine 2-Space and the Jacobian Conjecture

We study Lie subalgebras $L$ of the vector fields $\mathrm{Vec}^{c}({\mathbb A}^{2})$ of affine 2-space ${\mathbb A}^{2}$ of constant divergence, and we classify those $L$ which are isomorphic to the Lie algebra $\mathfrak{aff}_{2}$ of the group $\mathrm{Aff}_{2}(K)$ of affine transformations of ${\mathbb A}^{2}$. We then show that the following three statements are equivalent:
(a) The Jacobian Conjecture holds in dimension 2;
(b) All Lie subalgebras $L \subset \mathrm{Vec}^{c}({\mathbb A}^{2})$ isomorphic to $\mathfrak{aff}_{2}$ are conjugate under $\mathrm{Aut}({\mathbb A}^{2})$;
(c) All Lie subalgebras $L \subset \mathrm{Vec}^{c}({\mathbb A}^{2})$ isomorphic to $\mathfrak{aff}_{2}$ are algebraic.
Finally, we use these results to show that the automorphism groups of the Lie algebras $\mathrm{Vec}({\mathbb A}^{2})$, $\mathrm{Vec}^{0}({\mathbb A}^{2})$ and $\mathrm{Vec}^{c}({\mathbb A}^{2})$ are all isomorphic to $\mathrm{Aut}({\mathbb A}^{2})$.