In 2 dimensions, conformal Killing requires in index notation
$$ \nabla_a X_b + \nabla_b X_a = \nabla^c X_c g_{ab} $$
Taking the divergence with respect to the index $b$, you find
$$ \Delta_g X_a + \mathrm{Ric}_{ab}X^b = 0 \label{1}\tag{1}$$

Taking the product against $X$ and integrating over the whole manifold you find
$$ \int_M - \|\nabla X\|^2 + \mathrm{Ric}(X,X) ~\mathrm{dvol} = 0 $$
As a consequence you have the following generalization of Yano's theorem in reference 1 (also see remark below).

**Theorem** If $(M,g)$ is a two dimensional compact Riemannian manifold with non-positive Ricci curvature, and $\xi$ a conformal Killing field, then $\xi$ is parallel; furthermore, if $M$ is not Ricci-flat, then $\xi \equiv 0$.

The Theorem above requires that the Ricci curvature be signed; in two dimensions, this is equivalent to having the scalar curvature be signed. Luckily for us, we can make use of the Kazdan-Warner Theorems (see 2 below) which, among other things, state that in non-positive Euler characteristic, every metric is conformally equivalent to one that has constant curvature. As the condition that $\xi$ is a conformal Killing field is conformally invariant, this shows that

**Theorem** On a closed two dimensional Riemannian manifold with genus $\geq 1$, (a) all conformal Killing vector fields are Killing (in fact parallel) and (b) the conformal Killing field can only be non-trivial when genus $=1$.

**Remark** That equation \eqref{1} holds for Killing vector fields is one of the main tricks of the Bochner/Yano technique for studying vector fields on manifolds (an analogous trick can also be used to study harmonic vector fields). One of the many coincidences for studying the conformal geometry in 2 dimensions is that equation \eqref{1} also holds for conformal Killing fields in 2D (it does not hold in general in higher dimensions). Yano proved his theorem (ref. 1 below) for Killing fields in general dimensions; in the above we just note that the same argument can be applied also for conformal Killing fields in 2D.

*Yano, Kentaro*, **On harmonic and Killing vector fields**, Ann. Math. (2) 55, 38-45 (1952). ZBL0046.15603.
*Kazdan, Jerry L.; Warner, Frank W.*, **Curvature functions for compact 2-manifolds**, Ann. Math. (2) 99, 14-47 (1974). ZBL0273.53034.