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We expand a study of relationship between the Li content and activity of G and K dwarfs. In addition to high-active stars from Mishenina et al. (2012), we analyse the echelle spectra obtained with the ELODIE spectrograph of 127 stars with the lower activity level close to the solar one. An analysis of a full homogeneous data set of 280 stars shows a definite correlation between \log A(Li) and the chromosphere activity. Beside a common conclusion that the high Li content is typical in fast rotating stars with activity saturation, we found high Li abundances in a group of slower rotating stars (P_{rot} >10 days). Activity of all these stars depends only weakly on the rotation period, while stars with the low \log A(Li) and solar-type activity follow the law L_X ~ P^{-2}_{rot}. To understand this result, we re-analysed phenomenon of bimodal distribution of a number of G and K main-sequence stars versus an axial rotation for using the X-ray data, our estimates of a spot area after the Kepler observations, and frequences of occurrence of superflares as well. We show that sets of stars with the high Li content and having superflares posses general properties. We discuss a role of the dynamo mechanism in formation of activity and argue that for explanation of activity processes on superflare stars with the high Li abundance, it is necessary to take into account the initial conditions with which these G and K stars come to the main sequence.