  A combined analysis of data on isoscalar S-wave processes
$\pi\pi\to\pi\pi,K\overline{K},\eta\eta$ and on decays
$J/\psi\to\phi\pi\pi,\phi K\overline{K}$ from the DM2, Mark III and BESIII
collaborations is performed to study $f_0$ mesons. The method of analysis is
based on analyticity and unitarity and uses an uniformization procedure. In the
analysis limited only to the multi-channel $\pi\pi$-scattering data, two
possible sets of parameters of the $f_0(500)$ were found: in both cases the
mass was about 700 MeV but the total width was either about 600 or 930 MeV. The
extension of the analysis using only the DM2 and Mark III data on the $J/\psi$
decays does not allow to choose between these sets. However, the data from
BESIII on the di-pion mass distribution in the decay $J/\psi\to\phi\pi^+\pi^-$
clearly prefers the wider $f_0(500)$ state. Spectroscopic implications from
results of the analysis are also discussed.
