Classification of rotational surfaces in Euclidean space satisfying a linear relation between their principal curvatures

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We classify all rotational surfaces in Euclidean space whose principal curvatures κ₁ and κ₂ satisfy the linear relation κ₁=aκ₂+b, where a and b are two constants. We give a variational characterization of these surfaces in terms of its generating curve. As a consequence of our classification, we find closed (embedded and not embedded) surfaces and periodic (embedded and not embedded) surfaces with a geometric behaviour similar to Delaunay surfaces.