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We study Lie subalgebras $L$ of the vector fields $\mathrm{Vec}^{c}({\mathbbA}^{2})$ of affine 2-space ${\mathbb A}^{2}$ of constant divergence, and weclassify those $L$ which are isomorphic to the Lie algebra $\mathfrak{aff}_{2}$of the group $\mathrm{Aff}_{2}(K)$ of affine transformations of ${\mathbbA}^{2}$. We then show that the following three statements are equivalent: (i)The Jacobian Conjecture holds in dimension 2; (ii) All Lie subalgebras $L\subset \mathrm{Vec}^{c}({\mathbb A}^{2})$ isomorphic to $\mathfrak{aff}_{2}$are conjugate under $\mathrm{Aut}({\mathbb A}^{2})$; (iii) All Lie subalgebras$L \subset \mathrm{Vec}^{c}({\mathbb A}^{2})$ isomorphic to$\mathfrak{aff}_{2}$ are algebraic.

Lie Subalgebras of vector fields and the Jacobian Conjecture