Surfaces with straight lines as affine surfaces of centres (Q1841869)

It is well-known that a cylinder of revolution in Euclidean 3-space may be characterized by the property that all normals intersect two straight lines (namely the axis of the cylinder and the common improper line of each orthogonal plane to this axis). The author investigates the corresponding surfaces \(\Phi\) of class \(C^3\) without parabolic points in the equiaffine threespace where the focal surfaces of the affine normals degenerate either into two different straight lines \(g\) and \(h\) or into one straight line \(g\). In the first case \(\Phi\) must be an elliptic resp. hyperbolic affine surface of revolution with the axis \(g\), ellipses resp. hyperbolas as parallels in planes with the improper line \(h\) and meridians which may be computed explicitly. In the second case \(\Phi\) must be a conoidal ruled surface with generators in planes with the improper line \(g\). Vice versa the indicated surfaces have affine normals with either \(g\) and \(h\) or with \(g\) as degenerated focal surfaces.