  Let $M$ be a K\"ahler manifold with complex structure $J$ and K\"ahler metric
$g$. A c-projective vector field is a vector field on $M$ whose flow sends
$J$-planar curves to $J$-planar curves, where $J$-planar curves are analogs of
what (unparametrised) geodesics are for pseudo-Riemannian manifolds (without
complex structure). The c-projective symmetry algebras of K\"ahler surfaces
with essential (i.e., non-affine) c-projective vector fields are computed.
