We show that the second greatest possible dimension of the group of (local) almost isometrics of a Finsler metric is n(2)-n/2 + 1 for n = dim(M) not equal 4 and n(2)-n/2 + 2 = 8 for n = 4. If a Finsler metric has the group of almost isometrics of dimension greater than n(2)-n/2 + 1, then the Finsler metric is Randers, i.e., F(x, y) = root g(x) (y, y) + tau(y). Moreover, if n not equal 4, the Riemannian metric g has constant sectional curvature and, if in addition n not equal 2, the 1-form tau is closed, so (locally) the metric admits the group of local isometrics of the maximal dimension n(n+1)/2.
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