# Lie correspondence for Ind groups

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 $$G=H?$$