  Projective vector fields are the infinitesimal transformations whose local
flow preserves geodesics up to reparametrisation. In 1882 Sophus Lie posed the
problem of describing 2-dimensional metrics admitting a non-trivial projective
vector field, which was solved in recent years. In the present paper, we solve
the analog of Lie's problem in dimension 3, for Riemannian metrics and, more
generally, for Levi-Civita metrics of arbitrary signature.
