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A new analysis of S-wave production amplitudes for the reaction $\pi^-p_{\uparrow} \rightarrow \pi^+ \pi^- n$ on a transversely polarized target isperformed. It is based on the results obtained by the CERN-Cracow -Munichcollaboration in the pion-pion energy range from 600 MeV to 1600 MeV at 17.2GeV/c $\pi^-$ momentum. Energy-independent separation of the S-wavepseudoscalar amplitude ($\pi$ exchange) from the pseudovector amplitude($a_{1}$ exchange) is carried out using assumptions much weaker than those inall previous analyses. We show that, especially around 1000 MeV and around 1500MeV, the $a_{1}$ exchange amplitude cannot be neglected. The scalar-isoscalar$\pi\pi$ phase shifts are calculated using fairly weak assumptions. Below the$K\overline{K}$ threshold we find two solutions for the $\pi-\pi$ phase shifts,for which the phases increase slower with the effective $\pi-\pi$ mass than theP-wave phases. Both solutions are consistent with a broad $f_{0}(500)$ but onlyone is similar to the well-known "down" solution. We find also the thirdsolution (with a somewhat puzzling behavior of inelasticity) which exhibits anarrow $f_{0}(750)$ claimed by Svec. All the solutions undergo a rapid changeat the $K\overline{K}$ threshold. Above 1420 MeV the phase shifts increase withenergy faster than those obtained without the polarized-target data. This phasebehavior as well as an increase of the modulus of the $a_{1}$-exchangeamplitude can be due to the presence of the $f_{0}(1500)$.Comment: 25 Latex pages + 15 Postscript figure

Separation of $S$ -wave pseudoscalar and pseudovector amplitudes in $\pi^- p_{\uparrow} \rightarrow \pi^+ \pi^- n$ reaction on polarized target