My favorite 2D metrics of nonconstant curvature admitting a Killing vector field are the so-called Darboux-superintegrable metrics. Their definition is: the space of Killing tensors of degree two (i.e., integrals of the geodesic flow that are quadratic in velocities) is 4-dimensional. These metrics necessary admit Killing vector field. Their local description was known to Koenigs (Note II from Darboux "Sur la theorie generale des surfaces, Vol. IV, 1896) and S. Lie (1882), one can get a list of these metrics from arXiv:math-ph/0307039v1 or, alternatively from the paper arXiv:0705.3592 (the metrics 2a,2b,2c from Theorem 1). The lists are equivalent modulo coordinate transformation and contain also semi-riemannian metrics.
Let me explain why I love these metrics:
(1) They appear in differential geometry and physics. In differential geometry, they appear in studying of projective connections with big group of symmetries and their analysis was one of the main ingredients of the solution of two problems explicitly stated by S. Lie which is one of my bests results. They also appeared in mathoverflow: for example the metrics suggested in the Bryant's answer on Riemannian surfaces with an explicit distance function? are Darboux-superintegrable.
(2) They appear in physics: many physical phenomena lead to such metrics. There is a big activity about it in the "superintegrability" community (consult, for example, the papers of Winternitz et al in ArXive).
(3) For these metrics, one can answer many natural questions that normally require solving systems of ODE or PDE (such that description of geodesics or eigenvalues of the Laplacian) by algebraic methods, using the Killing tensors.
My personal opinion is that the metrics appeared naturally in differential geometry, because they are "the third best" metrics: the best metric is flat, the second best is of constant curvature, and the Darboux-superintegrable are the next simplest choice. The metrics appeared in physics, since physicists love solvable models, and models described by these metrics are sometimes solvable.
