13. Construct a surface of rotation having as its generators three given lines passing through one point and not lying in the same plane. How many solutions does the problem have? Do the resulting conical surfaces have common generators?
13. Let $S$ be the given point where the lines $A A^{\prime}, B B^{\prime}, C C^{\prime}$ intersect. From point $S$, six rays originate: $S A, S A^{\prime}, S B, S B^{\prime}$, $S C$ and $S C^{\prime}$. For each triplet of these rays, for example $S A, S B$, $S C$, a line can be constructed, the points of which are equidistant from these rays (see the solution to problem 5 in § 6). This line is the axis of a conical surface, the generators of which are the rays $S A$, $S B, S C$. Four such conical surfaces can be constructed (if we consider centrally symmetric conical surfaces with a common vertex and axis to be the same). Each of the given lines is a common generator of all four conical surfaces.