  An analytic closed form solution is derived for the bound states of electrons
subject to a static, inhomogeneous ($1/r$-decaying) magnetic field, including
the Zeeman interaction. The solution provides access to many-body properties of
a two-dimensional, non-interacting, electron gas in the thermodynamic limit.
Radially distorted Landau levels can be identified as well as magnetic field
induced density and current oscillations close to the magnetic impurity. These
radially localised oscillations depend strongly on the coupling of the spin to
the magnetic field, which give rise to non-trivial spin currents. Moreover, the
Zeeman interaction introduces a lowest flat band for $E_F=0^+$ assuming a spin
$g_s$-factor of two. Surprisingly, in this case the charge and current
densities can be computed analytically in the thermodynamic limit. Numerical
calculations show that the total magnetic response of the electron gas remains
diamagnetic (similar to Landau levels) independent of the Fermi energy.
However, the contribution of certain, infinitely degenerate energy levels may
become paramagnetic. Furthermore, numerical computations of the Hall
conductivity reveal asymptotic properties of the electron gas, which are driven
by the anisotropy of the vector potential instead of the magnetic field, i.e.
become independent of spin. Eventually, the distorted Landau levels give rise
to different Hall conductivity phases, which are characterized by sharp sign
flips at specific Fermi energies. Overall, our work merges "impurity" with
Landau-level physics, which provides novel physical insights, not only locally,
but also in the asymptotic limit. This paves the way for a large number of
future theoretical as well as experimental investigations.
