  Let $m=2l$ be a positive natural number, $l=1, 2, \ldots. $ A Finslerian
metric $F$ is called an $m$-th root metric if its $m$-th power $F^m$ is of
class $C^{m}$ on the tangent manifold $TM$. Using some homogenity properties,
the local expression of an $m$-th root metric is a polynomial of degree $m$ in
the variables $y^1$, $\ldots$, $y^n$, where $\dim M=n$. $F$ is locally
symmetric if each point has a coordinate neighbourhood such that $F^m$ is a
symmetric polynomial of degree $m$ in the variables $y^1$, $\ldots$, $y^n$ of
the induced coordinate system on the tangent manifold. Using the fundamental
theorem of symmetric polynomials, the reduction of the number of the
coefficients depending on the position makes the computational processes more
effective and simple. In the paper we present some general observations about
locally symmetric $m$-th root metrics. Especially, we are interested in
generalized Berwald surfaces with locally symmetric fourth root metrics. The
main result (Theorem 1) is their intrinsic characterization in terms of the
basic notions of linear algebra. We present a one-parameter family of examples
as well. The last section contains some computations in 3D. They are supported
by the MAPLE mathematics softwer (LinearAlgebra).
