Ruled and quadric surfaces in the 3-dimensional Euclidean space satisfying $Δ^{III}\boldsymbol{x} = \varLambda \boldsymbol{x}$

We consider ruled and quadric surfaces in the 3-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form $III$, i.e., their position vector $\boldsymbol{x}$ satisfies the relation $Δ^{III}\boldsymbol{x}=\varLambda \boldsymbol{x}$ where $\varLambda$ is a square matrix of order 3. We show that helicoids and spheres are the only surfaces in $E^3$ satisfying the preceding relation.

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