On submaximal dimension of the group of almost isometries of Finsler metrics

We show that the second greatest possible dimension of the group of (local) almost isometries of a Finsler metric is $\frac{n^2-n}{2}+1$ for $n= dim(M)\ne 4 $and$\frac{n^2 -n}{2} +2 =8$for$n=4$. If a Finsler metric has the group of almost isometries of dimension greater than $\frac{n^2-n}{2}+1$, then the Finsler metric is Randers, i.e., $F(x,y)=\sqrt{g_x(y,y)}+\tau (y)$. Moreover, if $n\mathrm{\ne }4$, the Riemannian metric $g$ has constant sectional curvature and, if in addition $n\mathrm{\ne }2$, the 1-form $\tau$ is closed, so (locally) the metric admits the group of local isometries of the maximal dimension $\frac{n(n+1)}{2}$.