A Killing field is preserved by the Ricci flow. By a theorem of Daskalopoulos, Hamilton and Sesum (arXiv:0902.1158), on a compact surface an ancient (defined for all negative time) solution to the Ricci flow which is not a shrinking soliton is diffeomorphism equivalent to the Fateev-Onofri-Zamolodchikov-King-Rosenau one-paremeter of metrics. These metrics have the form $$ g(t) = \frac{-4\sinh(2t)(dx^{2} + dy^{2})}{1 + 2\cosh(2t) r^{2} + r^{4}}$$ in which $z = x + iy$ is a standard coordinate on the complement of a point in the complex projective line, and $r = \sqrt{x^{2} + y^{2}}$. There is an obvious rotational symmetry. The scalar curvature $R(t)$ of $g(t)$ satisfies the bounds $$\frac{-2}{\sinh(2t)} = \min_{S^{2}}R(t) \leq R(t) \leq \max_{S^{2}}R(t) = -2\coth(2t).$$
For a rotationally symmetric metric on the sphere the Ricci flow can be rewritten as the logarithmic diffusion equation $u_{t} = (\log u)_{zz}$ (not the same $z$ as above). P. Rosenau and J.R. King independently found the above solution in the context of such diffusion equations. However, Fateev-Onofri-Zamolodchikov found this metric, which they called the sausage metric, earlier, in the context of studying the renormalization group flow for a two-dimensional sigma model, in their paper Integrable deformations of the O(3) sigma model. The sausage model, Nucl. Phys. B 406 (3), 521-565 (1993). (the Ricci flow is the one-loop renormalization group flow).
On a compact oriented surface of genus at least 2 there is no non-zero Killing field, as follows, for instance, from the classical Bochner argument. Similarly, on a torus, a non-zero Killing field must be parallel. On $S^{2}$ a Killing field generates an isometric $S^1$ action fixing two distinct points; that is, the metric is rotationally symmetric (a proof can be found in the paper of Chen, Lu, and Tian). What this shows is that in the compact case an answer to your question has to be on the sphere or torus. Since the Ricci flow preserves isometries, it is natural to ask what the are interesting rotationally symmetric Ricci flows (A recent survey is arXiv:1103.4669); the theorem above about ancient solutions is one sort of answer.