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The class of Bryant-type metrics is a  natural extension of the class of 4-th root Finsler metrics which are used in Biology as ecological metrics. In this paper, we  classify  Bryant-type metrics admitting an $(alpha, beta)$-metric on a two-dimensional  manifold and show that it contains two classes of non-Riemannian $(alpha, beta)$-metrics, specially Randers-type metrics. This might provide fine insights into a possible theory of deformations of Finsler norms.