  The universal variational expansion for the non-relativistic three-body
systems is explicitly constructed. Three-body universal expansion can be used
to perform highly accurate numerical computations of the bound state spectra in
arbitrary three-body systems, including Coulomb three-body systems with
arbitrary particle masses and electric charges. Our main interest is related to
the adiabatic three-body systems which contain one bound electron and two heavy
nuclei of hydrogen isotopes: the protium $p$, deuterium $d$ and tritium $t$. We
also consider the analogous (model) hydrogen ion ${}^{\infty}$H$^{+}_2$ with
the two infinitely heavy nuclei.
  PACS number(s): 36.10.-k and 36.10.Dr
