  In the previous paper by Pereira and the author, it was proved that any
pencil of plane curves of degree greater than one with irreducible generic
fiber can have at most five completely reducible fibers although no examples
with five such fibers were ever found. Recently Janis Stipins has proved that
if any two fibers of a pencil intersect transversally then it cannot have five
completely reducible fibers. In this paper we generalize the Stipins result to
arbitrary pencils. We also include into consideration more general special
fibers that are the unions of lines and non-reduced curves. These fibers are
important for characteristic varieties of line complements.
