  We give a complete list of mutually non-diffeomorphic normal forms for the
two-dimensional metrics that admit one essential (i.e., non-homothetic)
projective vector field. This revises a result from the literature and extends
the results of two papers, by R.L. Bryant & G. Manno & V.S. Matveev (2008) and
V.S. Matveev (2012) respectively, solving a problem posed by Sophus Lie in
1882.
