Energylevels are said to be degenerate, if the same energylevel is obtained by more than one quantum mechanical state.
The energy of its state is mainly determined by the electrostatic interaction of the (negative) electron with the (positive) nucleus, calculatable using the principal quantum number
Fine structure arises from relativistic kinetic energy corrections, spin-orbit coupling (magnetic dipole interaction of the electron spin with the magnetic field due to the movement in the electric field of the nucleus) and the Darwin term (contact term interaction of s-shell electrons inside the nucleus).
If the symmetry is broken by a perturbation, such as applying an external electric field, this can change the energies of the states, causing energylevel splitting.