The influence of three-particle breakup singularities on s-wave meson production in nucleon-nucleon collisions is studied within the distorted wave Born approximation. This study is based on a simple scalar model for the twonucleon interaction and the production mechanism. An algorithm for the exact numerical treatment of the inherent three-body cuts, together with its straightforward implementation is presented. It is also shown that two oftenused approximations to avoid the calculation of the three-body breakup are not justified. The possible impact on pion production observables is discussed. PACS number(s): 25.40Qa,13.60Le
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