  An $(\alpha,\beta)$-metric is defined by a Riemannian metric $\alpha$ and
$1$-form $\beta$. In this paper, we study a known class of two-dimensional
$(\alpha,\beta)$-metrics of vanishing S-curvature. We determine the local
structure of those metrics and show that those metrics are Einsteinian
(equivalently, isotropic flag curvature) but generally are not Ricci-flat.
