For a given (n ? 1)-dimensional hypersurface x : M → R n , consider the Laguerre form Φ, the Laguerre tensor L and the Laguerre second fundamental form B of the immersion x. In this article, we address the case when the Laguerre form of x is parallel, i.e., ?Φ ≡ 0. We prove that ?Φ ≡ 0 is equivalent to Φ ≡ 0, provided that either L+λB+μg = 0 for some smooth function λ and μ, or x has constant Laguerre eigenvalues, or x has constant para-Laguerre eigenvalues, where ? is the Levi-Civita connection of the Laguerre metric g
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