Holomorphic Lie Group Actions on Danielewski Surfaces

We prove that any Lie subgroup $G$ (with finitely many connected components) of an infinite-dimensional topological group $\mathcal{G}$ which is an amalgamated product of two closed subgroups, can be conjugated to one factor. We apply this result to classify Lie group actions on Danielewski surfaces by elements of the overshear group (up to conjugation).