  A curve C in the projective plane is called non-negative if the
self-intersection number of C after the minimal resolution of singularities of
C is non-negative. Given a unicuspidal rational plane curve C with singular
point P, we study the unique pencil Lambda_C on the projective plane satisfying
C is in Lambda_C and P is its unique base point. We show that the general
member of Lambda_C is a rational curve if and only if the curve C is
non-negative. We also show that in such a case then Lambda_C has a dicritical
of degree 1. Note that all currently known unicuspidal rational curves C in the
projective plane are non-negative.
