Separable representations of the S and P wave amplitudes are presented for the pion incident energy-range in the laboratory system: TL=0-400 MeV. We make separable potentials in rank one or two form and in one or multi-channel form, depending on the eigenstates. The relationship between the form factor of Pn potential and the 11NN vertex form factor in the one-pion-exchange potential is seriously taken into account. The potentials are made by fitting the two empirical partial wave amplitudes by Koch and Pietarinen and by Arndt, Ford and Roper. in a separable form with respect to the initial and final relative momenta. This makes the three-body calculation considerably tractable and transparent. Therefore the separable form of the potentials are very significant for the study of· the TCNN system. So far the separable TCN potentials have been presented in fitting the TCN phase shifts by various groups. 4 >.s> Among all, the Pu potential is relatively complicated. This eigenstate has the nucleon pole as well as the Roper resonance and requires the separable representation in a rank two form at least. Furthermore the form factor of the separable potential should relate with the TCNN vertex form factor appearing in the one-pion-exchange potential for nuclear forces. So far, however, this interrela­ tion is not so seriously taken into account. In this paper we put the strong condition that the form factor for the nucleon pole part of the Pu potential is the same with the TCNN vertex form factor for the one-pion-exchange potential. This should provide more realistic condition for the off-shell structure of the potential than the so far presented versions. The form factor of the TCNN vertex has been studied by various groups, since Ueda and Green introduced it into the one-boson-exchange potential in 1967. 6 > They used the size parameter A in the dipole parametrization as large as A=2-3 GeV. Although Bonn group made the search of a smaller value to be consistent with the values from subhadronic models: A~0.7-0.8 GeV, they find A~1.3 GeV. 7 > More recently Ueda has proposed that the one-pion-exchange with the TCNN form factor with A=2-3 GeV is interpreted as the sum of the one-pion-exchange with a soft form factor of A~0.9 GeV and the 1=1, r=o- part of the correlated three-pion­ exchange.2>.s>-lo) The latter is effectively represented by an isovector and pseudo­