  The Hamiltonian of a pure hydrogen atom possesses the SO(4) symmetry group
generated by the integrals of motion: the angular momentum and the Runge-Lenz
vector. The pure hydrogen atom is a supersymmetric and superintegrable system,
since the Hamilton-Jacobi and the Schr\"odinger equations are separable in
several different coordinate systems and has an exact analytical solution. The
Schr\"odinger equation for a hydrogen atom in a uniform electric field (Stark
effect) is separable in parabolic coordinates. The system has two conserved
quantities: z-projections of the generalized Runge-Lenz vector and of the
angular momentum. The problem is integrable and has the symmetry group
SO(2)xSO(2). The ion of the hydrogen molecule (problem of two Coulomb centers)
has similar symmetry group SO(2)xSO(2) generated by two conserved z-projections
of the generalized Runge-Lenz and of the angular momentum on the internuclear
axis. The corresponding Schr\"odinger equation is separable in the elliptical
coordinates. For the hydrogen atom in a uniform magnetic field, the respective
Schr\"odinger equation is not separable. The problem is non-separable and
non-integrable and is considered as a representative example of quantum chaos
that cannot be solved by any analytical method. Nevertheless, an exact
analytical solution describing the quantum states of a hydrogen atom in a
uniform magnetic field can be obtained as a convergent power series in two
variables, the radius and the sine of the polar angle. The energy levels and
wave functions for the ground and excited states can be calculated exactly,
with any desired accuracy, for an arbitrary strength of the magnetic field.
Therefore, the problem can be considered superintegrable, although it does not
possess supersymmetry and additional integrals of motion.
