  Within the framework of the generalized linear sigma model with glueballs
recently proposed [1,2], we study the schematic spectroscopy of scalar and
pseudoscalar mesons in the SU(3) flavor limit and explore their quark and glue
contents. In this framework, for both scalars and pseudoscalars, the two octet
physical states are admixtures of quark-antiquark and four-quark components,
and the three singlet states contain quark-antiquark, four-quark and glue
components. We identify the two scalar octets with $a_0(980)$ and $a_0(1450)$
and the two pseudoscalar octets with $\pi(137)$ and $\pi(1300)$. We show that,
as expected, the light pseudoscalar octet is made dominantly of
quark-antiquarks whereas the light scalar octet has a reversed substructure
with a dominant four-quark component. The case of singlets is more complex due
to surplus of states up to around 2 GeV. We consider all 35 permutations for
identifying the three pseudoscalar singlets of our model with three of the
seven experimental candidates. Our numerical simulation unambiguously
identifies the lightest and the heaviest pseudoscalar singlets with $\eta(547)$
and $\eta(2225)$, respectively, and favors the identification of the middle
singlet with either $\eta(1295)$ or $\eta(1405)$ [or, to a lesser extent, with
$\eta(1475)$] and thereby allows a probe of their substructures. We then
estimate the quark and glue components and find that the three pseudoscalar
singlets (from lightest to heaviest) are mainly of quark-antiquark, four-quark
and glue substructure, while the corresponding three scalar singlets (from
lightest to heaviest) are of four-quark, quark antiquark and glue contents. The
masses of pure pseudoscalar and scalar glueballs are estimated around 2.0 and
1.6 GeV, respectively.
