  The generalized dual-kinetic-balance approach for axially symmetric systems
is employed to solve the two-center Dirac problem. The spectra of one-electron
homonuclear quasimolecules are calculated and compared with the previous
calculations. The analysis of the monopole approximation with two different
choices of the origin is performed. Special attention is paid to the lead and
xenon dimers, Pb$^{82+}$-Pb$^{82+}$-e$^{-}$ and Xe$^{54+}$-Xe$^{54+}$-e$^{-}$,
where the energies of the ground and several excited $\sigma$-states are
presented in the wide range of internuclear distances. The developed method
provides the quasicomplete finite basis set and allows for construction of the
perturbation theory, including within the bound-state QED.
