We study hypersurfaces in Euclidean space ℝn+1 whose position vector x satisfies the condition L k x = Ax + b, where L k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k=0,...n-1, A ∈ (n+1) ℝ(n+1)×(n+1) is a constant matrix and b ∈ ℝ n+1 is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form double struck S signm (r) × ℝn-m with k + 1 < m < n-1. This extends a previous classification for hypersurfaces in ℝn+1 satisfying Δx = Ax+b, where Δ = L0 is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A 53, 377-384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117-129 (1991)]. © Springer Science + Business Media B.V. 2007.