  The $\Lambda p \to \Lambda p \pi^0$ reaction near threshold is studied within
an effective Lagrangian method. The production process is described by
single-pion and single-kaon exchange. In addition to the role played by the
$\Sigma^*(1385)$ resonance of spin-parity $J^P = 3/2^+$, the effects of a newly
proposed $\Sigma^*$ ($J^P = 1/2^-$) state with mass and width around $1380$ MeV
and $120$ MeV are investigated. We show that our model leads to a good
description of the experimental data on the total cross section of the $\Lambda
p \to \Lambda p \pi^0$ reaction by including the contributions from the
possible $\Sigma^*(\frac{1}{2}^-)$ state. However, the theoretical calculations
by considering only the $\Sigma^*(1385)$ resonance fail to reproduce the
experimental data, especially for the enhancement close to the reaction
threshold. On the other hand, it is found that the single-pion exchange is
dominant. Furthermore, we also demonstrate that the angular distributions
provide direct information of this reaction, hence could be useful for the
investigation of the existence of the $\Sigma^*(\frac{1}{2}^-)$ state and may
be tested by future experiments.
