On a class of generalized Berwald manifolds

Let $F=\alpha\phi(s)$, $s:=\beta/\alpha$, be a generalized Berwald $(\alpha,\beta)$-metric on a $2$-dimensional manifold. Suppose that $F$ has vanishing $S$-curvature and $\phi'(0)\neq 0$. We show that if $F$ is regular, then it is a locally Minkowskian metric. If $F$ is an almost regular and non-locally Minkowskian metric, then we explicitly determine the function $\phi$ which results in generalized Berwald metrics not belonging to the classes of Berwald, Landsberg or Douglas metrics. Furthermore, we prove that a left-invariant Finsler metric on a $2$-dimensional Lie group has vanishing $S$-curvature if and only if it is a Riemannian metric of constant Gaussian curvature. Finally, we construct a family of Randers-type generalized Berwald metrics on an arbitrary odd-dimensional manifold.