The strangeness photoproduction processes near the threshold have been intensively studied by measuring K + in reactions such as γ + p → K + + Λ(σ0). There has been no reliable data on the neutron and as a consequence theoretical investigation suffered seriously from the lack of the data. We have an effort to measure the γ + d → K0(Λ) + X reaction in the π +π - (pπ -) decay channel of KS0 (Λ) using a liquid D2 target and internally-tagged photon beams (Eγ = 0.80-1.08GeV) at Research Center for Electron Photon Science (ELPH), Tohoku University. We have renewed the spectrometer (NKS2) and took data with a liquid D2 target in 2005-2007. More recently, we took data in 2010 after a detector upgrade in the vertex region to substantially increase the acceptance. The results of Λ single measurements on a deuterium target were obtained for differential cross-section as a function of momentum, as a function of angle, and integrated cross-section as a function of beam energy. The results are compared with recent theoretical studies: Isobar models (Kaon-MAID and Saclay-Lyon A) and a Regge-plus-Resonance model. © 2013 Elsevier B.V.