fact that the ISGMR in $^{208}$Pb does not constrain the compression modulus of symmetric nuclear matter but rather the one of neutron-rich matter, i.e., the isobaric incompressibility coefficient in Eq. (\[Ksat\]). From Eq. (\[Ksat\]) it is clear that the ISGMR in $^{208}$Pb (with an isospin asymmetry of $\delta =0.21$) should be sensitive to a linear combination of $K_{0}$ and $K_{\mathrm{sat,2}}$. The $K_{\mathrm{sat,2}}$ parameter is completely determined by the slope and curvature of the symmetry energy at saturation density as well as the third derivative of the EOS of symmetric nuclear matter (see, e.g., Ref. [@Che09]). Fig. [XKsat2]{} shows the $K_{\mathrm{sat,2}}$ parameter from SHF with MSL0 by varying individually $L$ and $m_{s,0}^{\ast }$. As can be seen in Fig. [XKsat2]{}, the $K_{\mathrm{sat,2}}$ parameter decreases with both $L$ and $m_{s,0}^{\ast }$, and thus $K_{A}(N,Z)$ for $^{208}$Pb will decrease correspondingly if the $K_{\mathrm{sat,2}}$ parameter has similar effects on $K_{A}(N,Z)$ as the $K_{\tau }$ parameter and the $K_{\mathrm{ss}}$ term as well as the other higher-order terms in Eq. (\[KA1\]) are not important for $K_{A}(N,Z)$. These results provide an explanation on the behavior that the ISGMR energies decrease with $L$ and $m_{s,0}^{\ast }$ observed in Fig. \[XEaPb208\].

![The $\protect\sigma $ parameter obtained from SHF with 