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In this paper, we study one of the open problems in Finsler geometry whichpresented by Matsumoto-Shimada about the existence of P-reducible metric whichis not C-reducible. For this aim, we study a class of Finsler metrics calledgeneralized P-reducible metrics that contains the class of P-reducible metrics.We prove that every generalized P-reducible $(\alpha, \beta)$-metric withvanishing S-curvature reduces to a Berwald metric or C-reducible metric. Itresults that there is not any concrete P-reducible $(\alpha,\beta)$-metric withvanishing S-curvature.

Generalized P-Reducible $(\alpha, \beta)$-Metrics with Vanishing S-curvature