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Abstract The local structure of rotationally symmetric Finsler surfaces with vanishing flag curvature is completely determined in this paper. A geometric method for constructing such surfaces is introduced. The construction begins with a planar vector field X that depends on two functions of one variable. It is shown that the flow of X could be used to generate a generalized Finsler surface with zero flag curvature. Moreover, this generalized structure reduces to a regular Finsler metric if and only if X has an isochronous center. By relating X to a Liénard system, we obtain the isochronicity condition and discover numerous new examples of complete flat Finsler surfaces, depending on an odd function and an even function.
Mu et al. (Mon,) studied this question.