Abstract In this paper, we study surfaces z= (x, y) z = φ (x, y) in Euclidean space that satisfy the equation ₗₗ+ ₘₘ= 2 φ xx + φ yy = Λ 2 where R Λ ∈ R is a real constant. We classify these surfaces when they are the zero level sets of an implicit equation of the type f (x) +g (y) +h (z) =0 f (x) + g (y) + h (z) = 0, where f, g and h are smooth functions of one variable. If =0 Λ = 0, we find a large family of surfaces with interesting symmetry properties. However, if =0 Λ ≠ 0, we show that the surfaces must be either surfaces of revolution or of the type z=f (x) +g (y) z = f (x) + g (y) ; furthermore, explicit parametrizations of these surfaces are obtained.
Rafael López (Mon,) studied this question.