On a class of two-dimensional Einstein Finsler metrics of vanishing S-curvature

An (α,β)-metric is defined by a Riemannian metric α and 1-form β. We have characterized a class of two-dimensional (α,β)-metrics of isotropic (thus vanishing) S-curvature. In this paper, we determine the local structure of those metrics and show that those metrics are Einsteinian (equivalently, of isotropic flag curvature) but not Ricci-flat in general.