For hypersurfaces with mean curvature H = ⟨η, v⟩ + λ, the paper gives explicit parametrizations of cylindrical flat examples and a complete classification of rotational examples.
Bueno, The Björling problem for prescribed mean curvature surfaces in 3 , Ann
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Invariant hypersurfaces with linear prescribed mean curvature
For hypersurfaces with mean curvature H = ⟨η, v⟩ + λ, the paper gives explicit parametrizations of cylindrical flat examples and a complete classification of rotational examples.