  Total energies and various bound state properties are determined for the
ground states in all six four-body muonic $a^{+} b^{+} \mu^{-} e^{-}$
quasi-atoms. These quasi-atoms contain two nuclei of the hydrogen isotopes
$p^{+}, d^{+}, t^{+}$, one negatively charged muon $\mu^{-}$ and one electron
$e^{-}$. In general, each of the four-body muonic $a^{+} b^{+} \mu^{-} e^{-}$
quasi-atoms, where $(a, b) = (p, d, t)$, can be considered as the regular
one-electron (hydrogen) atom with the complex nucleus $a^{+} b^{+} \mu^{-}$
which has a finite number of bound states. Furthermore, all properties of such
quasi-nuclei $a^{+} b^{+} \mu^{-}$ are determined from highly accurate
computations performed for the three-body muonic ions $a^{+} b^{+} \mu^{-}$
with the use of pure Coulomb interaction potentials between particles. It is
shown that the bound state spectra of such quasi-atoms are similar to the
spectrum of the regular hydrogen atoms, but there are a few important
differences. Such differences can be used in future experiments to improve the
overall accuracy of current evaluations of various properties of hydrogen-like
systems, including the lowest-order relativistic and QED corrections.
