Symmetry of a Hydrogen Atom in a Weak Magnetic Field

The degeneracy problem of a hydrogen atom in a uniform weak magnetic field $H$ is considered from the standpoint of the invariance group that the system obeys. Evidently the degeneracy space has a constant $z$ component of the angular momentum. Under such a classification, the dynamical invariance of the problem is shown to be an isomorph to the Lie group of the linear transformation of the straight line.