extensively investigated in the literature [@Pie02; @Vre03; @Agr03; @Pie04; @Col04]. It was firstly proposed by Piekarewicz [@Pie02] that the different symmetry energies used in the non-relativistic models and the relativistic models may be responsible for the puzzle that the former predicted an incompressibility in the range of $K_{0}=210-230$ MeV while the latter predicted a significantly larger value of $K_{0}\approx 270$ MeV from the analysis of the ISGMR centroid energy. It is seen from Fig. \[XEaPb208\] that a larger $L$ value (as in usual relativistic models) leads to a smaller $E_{ave}$ value and thus a larger $K_{0}$ value is necessary to counteract the deceasing of $E_{ave}$ due to a larger $L$ value. Furthermore, Fig. \[XEaPb208\] shows that $E_{ave}$ displays a very weak dependence on $E_{\text{\textrm{sym}}}(\rho _{0})$, which is in contrast to the results in Ref. [@Col04] where $E_{ave}$ is shown to be sensitive to $E_{\text{\textrm{sym}}}(\rho _{0})$. This is due to the fact that a constrain on the value of $E_{\text{\textrm{sym}}}({\rho
=0.1}$ [fm]{}$^{{-3}})$ was imposed in Ref. [@Col04], which leads to a strong linear correlation between $E_{\text{\textrm{sym}}}({\rho _{0}})$ and $L$ as shown recently in Ref. [@Che11a].

The symmetry energy dependence of the ISGMR centroid energy of $^{208}$Pb can be understood from the 