Classification of 3-dimensional Landsbergian $(\alpha,\beta)$-metrics

In this paper, we classify the class of 3-dimensional $(\alpha,\beta)$-metrics with vanishing Landsberg curvature. More precisely, we show that these metrics belong to one of the following main classes: Berwald metrics which contain the Randers or Kropina metrics, or satisfy an ODE. Some of the well-known unicorns are special solutions of this ODE.