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