There is presented a unified analysis of all available data on the decays of bottomonia $\Upsilon(mS)\to\Upsilon(nS)\pi\pi$ ($m>n$, $m=2,3,4,5,$ $n=1,2,3$), charmonia $J/\psi\to\phi(\pi\pi, K\overline{K})$, $\psi(2S)\to J/\psi\pi\pi$ and the data on isoscalar $S$-wave processes $\pi\pi\to\pi\pi,K\overline{K},\eta\eta$. The multi-channel $\pi\pi$ scattering is described in our model-independent approach based on analyticity and unitarity and using an uniformization procedure. It is shown that the basic shape of dipion mass distributions in the two-pion transitions of both charmonia and bottomonia states are explained by an unified mechanism based on the contribution of the $\pi\pi$, $K\overline{K}$ and $\eta\eta$ coupled channels including their interference. The role of the individual $f_0$ resonances in contributing to the dipion mass distributions in indicated decays of these states is considered. Since the satisfactory description of these decays with allowing for the $\eta\eta$ channel (in addition to the $\pi\pi$ and $K\overline{K}$ one) did not require any change of the $f_0$-meson parameters, the results of the analysis confirm convincingly all of our earlier conclusions on the scalar mesons.

[1] Yu.S. Surovtsev et al., J. Phys. G: Nucl. Part. Phys. 41 (2014) 025006.

[2] Yu.S. Surovtsev et al., Phys. Rev. D 89 (2014) 036010.

[3] Yu.S. Surovtsev et al., Phys. Rev. D 91 (2015) 037901.

[4] Yu.S. Surovtsev et al., Phys. Rev. D 92 (2015) 036002.