  The nuclear symmetry energy $E_{sym}(\rho)$ and its density slope $L(\rho)$
can be decomposed analytically in terms of the single-nucleon potential in
isospin asymmetric nuclear matter. Using three popular nuclear effective
interaction models which have been extensively used in nuclear structure and
reaction studies, namely, the isospin and momentum dependent MDI interaction
model, the Skyrme Hartree-Fock approach and the Gogny Hartree-Fock approach, we
analyze the contribution of different terms in the single-nucleon potential to
the $E_{sym}(\rho)$ and $L(\rho)$. Our results show that the observed different
density behaviors of $E_{sym}(\rho)$ for different interactions are essentially
due to the variation of the symmetry potential $U_{sym,1}(\rho,k)$.
Furthermore, we find that the contribution of the second-order symmetry
potential $U_{sym,2}(\rho,k)$ to the $L(\rho)$ generally cannot be neglected.
Moreover, our results demonstrate that the magnitude of the $U_{sym,2}(\rho,k)$
is generally comparable with that of $U_{sym,1}(\rho,k)$, indicating the
second-order symmetry potential $U_{sym,2}(\rho,k)$ may have significant
corrections to the widely used Lane approximation to the single-nucleon
potential in extremely neutron(proton)-rich nuclear matter.
