  Role of the $\Lambda(1600)$ is studied in the $K^- p \to \Lambda \pi^0 \pi^0$
reaction by using the effective Lagrangian approach near the threshold. We
perform a calculation for the total and differential cross sections by
considering the contributions from the $\Lambda(1600)$ and $\Lambda(1670)$
intermediate resonances decaying into $\pi^0 \Sigma^{*0}(1385)$ with
$\Sigma^{*0}(1385)$ decaying into $\pi^0 \Lambda$. Besides, the non-resonance
process from $u$-channel nucleon pole is also taken into account. With our
model parameters, the current experimental data on the total cross sections of
the $K^- p \to \Lambda \pi^0 \pi^0$ reaction can be well reproduced. It is
shown that we really need the contribution from the $\Lambda(1600)$ with
spin-parity $J^P = 1/2^+$, and that these measurements can be used to determine
some of the properties of the $\Lambda(1600)$ resonance. Furthermore, we also
plot the $\pi^0 \Lambda$ invariant mass distributions which could be tested by
the future experimental measurements.
