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New calculations of the quasi-bound state in the $K^- pp$ system usingFaddeev-type equations in AGS form with coupled $\bar{K}NN$ and $\pi \Sigma N$channels were performed. A chiral $\bar{K}N$ potential together withphenomenological models of $\bar{K}N$ interaction with one- and two-polestructure of the $\Lambda(1405)$ resonance were used. All three potentialsreproduce experimental data on low-energy $K^- p$ scattering and kaonichydrogen with the same level of accuracy. New method of calculating thesubthreshold resonance position and width in a three-body system was proposedand used together with the direct search of the resonance pole. We obtainedbinding energy of the $K^-pp$ quasi-bound state $\sim 32$ MeV with the chirallymotivated and $47 - 54$ MeV with the phenomenological $\bar{K}N$ potentials.The width is about $50$ MeV for the two-pole models of the interaction, whilethe one-pole potential gives $\sim 65$ MeV width. The question of using anenergy dependent potential in few-body calculations is discussed in detail. Itis shown that ``self-consistent'' variational calculations using such apotential are unable to produce a reasonable approximation to the exact result.

Faddeev calculations of the $\bar{K}NN$ system with chirally-motivated $\bar{K}N$ interaction. II. The $K^- pp$ quasi-bound state