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