  Solutions of the Dirac equation for an electron in the Coulomb potential are
obtained using operator invariants of the equation, namely the Dirac,
Johnson-Lippmann and recently found new invariant. It is demonstrated that
these operators are the spin invariants. The generalized invariant is
constructed and the exact general solution of the Dirac equation are found. In
particular, the explicit expressions of the bispinors corresponding to the
three complete sets of the invariants, their eigenvalues and quantum numbers
are calculated. It is shown that the general solution of one center Coulomb
Dirac equation contains free parameters. Changing one or more of these
parameters, one can transform one solution of the Dirac equation into any
other. It is shown for the first time that these invariants determine electron
spatial probability amplitude and spin polarization in each quantum state.
Electron probability densities and spin polarizations are explicitly calculated
in the general form for several electron states in the hydrogen-like energy
spectrum. Spatial distributions of these characteristics are shown to depend
essentially on the invariant set, demonstrating, in spite of the accidental
degeneracy of energy levels, physical difference of the states corresponding to
different spin invariants.
