The rare decay K0(L) --> pi0 pi0 nu nu-bar was studied with the E391a detector at the KEK 12-GeV proton synchrotron. Based on 9.4 x 10^9 K0L decays, an upper limit of 8.1 x 10^{-7} was obtained for the branching fraction at 90% confidence level. We a
lso set a limit on the K0(L) --> pi0 pi0 X (X --> invisible particles) process; the limit on the branching fraction varied from 7.0 x 10^{-7} to 4.0 x 10^{-5} for the mass of X ranging from 50 MeV/c^2 to 200 MeV/c^2.
We performed a search for the decay $K_L^0 rightarrow 3gamma$ with the E391a detector at KEK. In the data accumulated in 2005, no event was observed in the signal region. Based on the assumption of $K_L^0 rightarrow 3gamma$ proceeding via parity-viol
ation, we obtained the single event sensitivity to be $(3.23pm0.14)times10^{-8}$, and set an upper limit on the branching ratio to be $7.4times10^{-8}$ at the 90% confidence level. This is a factor of 3.2 improvement compared to the previous results. The results of $K_L^0 rightarrow 3gamma$ proceeding via parity-conservation were also presented in this paper.
