  A linear connection on a Finsler manifold is called compatible to the Finsler
function if its parallel transports preserve the Finslerian length of tangent
vectors. Generalized Berwald manifolds are Finsler manifolds equipped with a
compatible linear connection. In the paper we present a general and intrinsic
method to characterize the compatible linear connections on a Finsler manifold
of dimension three. We prove that if a compatible linear connection is not
unique then the indicatrices must be Euclidean surfaces of revolution. The
surplus freedom of choosing compatible linear connections is related to
Euclidean symmetries. The unicity of the solution of the compatibility
equations can be provided by some additional requirements. Following the idea
in \cite{V14} we are also looking for the so-called extremal compatible linear
connection minimizing the norm of its torsion at each point of the manifold.
