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, Pb⁸²⁺–Pb⁸²⁺–e⁻ and Xe⁵⁴⁺–Xe⁵⁴⁺–e⁻, where the energies of the ground and several excited σ-states are presented in the wide range of internuclear distances. The developed method provides the quasicomplete finite basis set and allows for the construction of perturbation theory, including within the bound-state QED.