Complete $\lambda $-hypersurfaces with constant squared norm of the second fundamental form in the Euclidean space $\mathbb R^{4}$

Under the assumption that the quasi-Gauss–Kronecker curvature $K_{q}$ is identically zero, we give a complete classification of $3$-dimensional complete $\lambda $-hypersurfaces with constant squared norm $S$ of the second fundamental form in the Euclidean space $\mathbb R^{4}$, where $S=\sum_{i,j}h^{2}_{ij}$, $K_{q}= \mathrm{det}(h_{ij}-\frac{1}{3}H\delta _{ij})$, with $h_{ij}$ being the components of the second fundamental form.