Translation surfaces of coordinate finite type

We consider translation surfaces in the 3-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form $III$, i.e. their position vector $x$ satisfies the relation $Δ^{III}x = Λx$, where $Λ$ is a square matrix of order 3. We show that Sherk's minimal surface is the only translation surface satisfying $Δ^{III}x = Λx$.

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