  Recently several experimental groups analysed data on $\pi^- p \to \eta \pi^-
p$ and $\pi^- p \to \eta \pi^0 n$ reactions with exotic $1^{-+}$ $P$-wave and
found a conflicting evidence for an exotic meson $I=1 1^{-+} (1405)$. High
statistics data on these reactions are presently analysed by BNL E852
Collaboration. All these analyses are based on the crucial assumption that the
production amplitudes do not depend on nucleon spin. This assumption is in
sharp conflict with the results of measurements of $\pi^- p \to \pi^- \pi^+ n$,
$\pi^+ n \to \pi^+ \pi^- p$ and $K^+ n \to K^+ \pi^- p$ on polarized targets at
CERN which find a strong dependence of production amplitudes on nucleon spin.
To ascertain the existence of exotic meson $1^{-+} (1405)$, it is necessary to
perform a model-independent amplitude analysis of reactions $\pi^- p \to \eta
\pi^- p$ and $\pi^- p \to \eta\pi^0 n$. We demonstrate that measurements of
these reactions on transversely polarized targets enable the required model
independent amplitude analysis without the assumption that production
amplitudes are independent on nucleon spin. We suggest that high statistics
measurements of reactions $\pi^- p \to \eta\pi^- p$ and $\pi^- p \to\eta\pi^0
n$ be made on polarized targets at BNL and at Protvino IHEP, and that
model-independent amplitude analyses of this polarized data be performed to
advance hadron spectroscopy on the level of spin dependent production
amplitudes.
