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We extend the Fixed Center Approximation (FCA) to the Faddeev Equations forthe $\bar{K} NN$ system with S=0, including the charge exchange mechanisms inthe $\bar{K}$ rescattering which have been ignored in former works within theFCA. We obtain similar results to those found before, but the binding isreduced by 6 MeV. At the same time we also evaluate the explicit contributionthe $\pi N \Sigma$ intermediate state in the three body system and find that itproduces and additional small decrease in the binding of about 3 MeV. Thesystem appears bound by about 35 MeV and the width omitting two bodyabsorption, is about 50 MeV.

Improved Fixed Center Approximation of the Faddeev equations for the $\bar{K}NN$ system with S=0