  A new analysis of S-wave production amplitudes for the reaction $\pi^-
p_{\uparrow} \rightarrow \pi^+ \pi^- n$ on a transversely polarized target is
performed. It is based on the results obtained by the CERN-Cracow -Munich
collaboration in the pion-pion energy range from 600 MeV to 1600 MeV at 17.2
GeV/c $\pi^-$ momentum. Energy-independent separation of the S-wave
pseudoscalar amplitude ($\pi$ exchange) from the pseudovector amplitude
($a_{1}$ exchange) is carried out using assumptions much weaker than those in
all previous analyses. We show that, especially around 1000 MeV and around 1500
MeV, 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 the
P-wave phases. Both solutions are consistent with a broad $f_{0}(500)$ but only
one is similar to the well-known "down" solution. We find also the third
solution (with a somewhat puzzling behavior of inelasticity) which exhibits a
narrow $f_{0}(750)$ claimed by Svec. All the solutions undergo a rapid change
at the $K\overline{K}$ threshold. Above 1420 MeV the phase shifts increase with
energy faster than those obtained without the polarized-target data. This phase
behavior as well as an increase of the modulus of the $a_{1}$-exchange
amplitude can be due to the presence of the $f_{0}(1500)$.
