  In this paper, we investigate rotational hypersurfaces family in $n$
-dimensional Euclidean space ${\mathbb{E}}^{n}$. Our focus is on studying the
Gauss map $\mathcal{G}$ of this family with respect to the operator
$\mathbb{L}_{k}$, which acts on functions defined on the hypersurfaces. The
operator $\mathbb{L}_{k}$ can be viewed as a modified Laplacian and is known by
various names, including the Cheng--Yau operator in certain cases.
Specifically, we focus on the scenario where $k=n-3$ and $n\geq 3$. By applying
the operator $\mathbb{L}_{n-3}$ to the Gauss map $\mathcal{G}$, we establish a
classification theorem. This theorem establishes a connection between the
$n\times n$ matrix $\mathcal{A}$, and the Gauss map $\mathcal{G}$ through the
equation $\mathbb{L}_{n-3}\mathcal{G}=\mathcal{A}\mathcal{G}$.
