  The pseudoscalar mesons with the masses higher than 1 GeV are assumed to
belong to the meson decuplet including the glueball as the basis state
supplementing the standard $SU(3)_F$ nonet of light $q\bar{q}$ states
$(u,d,s)$. The decuplet is investigated by means of an algebraic approach based
on hypothesis of vanishing the exotic $SU(3)_F$ commutators of "charges" and
their time derivatives. These commutators result in a system of equations
determining contents of the isoscalar octet state in the physical isoscalar
mesons as well as the mass formula including all masses of the decuplet:
$\pi(1300)$, K(1460), $\eta(1295)$, $\eta(1405)$ and $\eta(1475)$. The physical
isoscalar mesons $\eta_i$, are expressed as superpositions of the "ideal"
$q\bar{q}$ states ($N$ and $S$) and the glueball $G$ with the mixing
coefficient matrix following from the exotic commutator restrictions. Among
four one-parameter families of the calculated mixing matrix (numerous solutions
result from bad quality of data on the $\pi(1300)$ and K(1460) masses) there is
one family attributing the glueball-dominant composition to the $\eta(1405)$
meson. Similarity between the pseudoscalar and scalar decuplets, analogy
between the whole spectra of the $0^{-+}$ and $0^{++}$ mesons and affinity of
the glueball with excited $q\bar{q}$ states are also noticed.
