  Given a curve C on a projective nonsingular rational surface S, over an
algebraically closed field of characteristic zero, we are interested in the set
Omega_C of linear systems Lambda on S satisfying C is in Lambda, dim Lambda >
0, and the general member of Lambda is a rational curve. The main result of the
paper gives a complete description of Omega_C and, in particular, characterizes
the curves C for which Omega_C is non empty.
