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I have the following question (with a negative answer in general).

Consider an Ind algebraic group (e.g. a group object in category of ind varieties) as defined by Shafrevich. It was assumed for some time that if a subgroup of a connected Ind group has the same Lie algebra as the Lie algebra of the group then it coincides with the group. A counterexample was given in Furter and Kraft in $\S$17.3. There, however, non-tame automorphisms of $\mathbb{A}^3$ played an important role.

My question is: does this "Lie correspondence" hold in dimension $2?$ In particular, is it true that if a Lie algebra $\mathcal{Lie}(G)$ of a connected closed subgroup $G$ of the group $\operatorname{Aut}(\mathbb{A}^2)$ coincides with an algebra $\mathcal{Lie}(H)$ of another connected closed subgoup $H<\operatorname{Aut}(\mathbb{A}^2)$ then $G=H?$

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