  Exhaustive ghost solutions to Einstein-Weyl equations for two dimensional
spacetimes are obtained, where the ghost neutrinos propagate in the background
spacetime, but do not influence the background spacetime due to the vanishing
stress-energy-momentum tensor for the ghost neutrinos. Especially, those
non-trivial ghost solutions provide a counterexample to the traditional claim
that the Einstein-Hilbert action has no meaningful two dimensional analogue.
