[en] An amplitude analysis of the K^-K⁺ system produced in the reactions π^-p → K^-K⁺n and π⁺n → K^-K⁺p was carried out using data from a high statistics experiment carried out with the Argonne Effective Mass Spectrometer. Combining the results from the two reactions allows the separation of the KantiK production amplitudes into their isospin-zero and one components. An enhancement is deserved in the S-wave amplitude near 1300 MeV which we call the S'. Using various physical arguments, including the t dependence of the amplitudes, six out of eight mathematically-allowed solutions can be eliminated for the magnitudes and phases of the amplitudes and shown that this new S-wave state has isospin zero. Requiring a reasonable behavior for the isospin-one P wave shows that the S' amplitude has a slow phase variation through the peak region