  Based on the fact that the Hamiltonians of the Coulomb many-particle systems
are always factorized we develop the two different approaches for analytical
solution of the Schr\"{o}dinger equation written for arbitrary few- and
many-particle Coulomb systems. The first approach is the matrix factorization
method. Another method is based on the $D^{+}-$series of representations of the
hyper-radial O(2,1)-algebra. The both these methods allow us to obtain the
closed analytical formulas for the bound state energies in an arbitrary
many-particle Coulomb system.
