We propose to study the production of light vector mesons $V=\rho,\omega, f(1270), K^*(892)$ off nuclei targets using the COMPASS-like detector and the M2 pion, kaon and muon beams at CERN\cite{AMBER18}. \\
The scattering amplitude of a vector meson off the nucleon at zero angle, averaged over the nucleon spin is determined by two quantities: $\sigma'_T=\sigma _T(1-i \alpha _T)$
and $\sigma'_L=\sigma _T(1-i\alpha _L)$, where $\sigma_{T(L)}$ is the total cross section for the interaction of a transversely (longitudinally) polarized vector meson with the nucleon and
$\alpha_{T(L)}= Re f_{T(L)}(0)/Im f_{T(L)}(0)$ is the ratio of the real to imaginary part of the corresponding amplitudes at zero angle. \\
Whereas in the charge exchange reaction $\pi^-+p\to V+n$ with dominating pion exchange vector mesons are produced mainly longitudinally polarized (helicity $\lambda=0$) the investigation of
their production off nuclei $\pi^-+ A \to V+ A^\prime$ ($A^\prime$ is the nuclear excitation or its break-up products) allows to extract the total cross section of the
longitudinally polarized vector meson interaction with nucleon $\sigma_L(V N)$, a value of which has not yet been measured.\\
Such measurements are of a current interest as they are complement to investigations of vector mesons photoproduction off nuclei~\cite{chudakov16} the experimental realization of which is recently proposed at JLAB~\cite{LOI15, proposal17} to be done using the possibilities of the GlueX detector at Hall D, JLAB.\\
The obvious advantage of the vector mesons production by hadron beam compared with vector mesons photoproduction is that in the charge exchange process $\pi^-(K^-)+A\to V+A'$
the main part of vector mesons is longitudinally polarized unlike vector mesons photoproduction, where due to the s-channel helicity conservation $\rho,\phi$ mesons are produced mainly transversely polarized and only a part of $\omega$ mesons at JLAB energies (due to pion exchange) is longitudinally polarized.