  The eigenvalues and eigenfunctions of a linear {\alpha}^{2}-dynamo have been
computed for different spatial distributions of an isotropic \alpha-effect.
Oscillatory solutions are obtained when \alpha exhibits a sign change in the
radial direction. The time-dependent solutions arise at so called exceptional
points where two stationary modes merge and continue as an oscillatory
eigenfunction with conjugate complex eigenvalues. The close proximity of
oscillatory and non-oscillatory solutions may serve as the basic ingredient for
reversal models that describe abrupt polarity switches of a dipole induced by
noise.
  Whereas the presence of an inner core with different magnetic diffusivity has
remarkable little impact on the character of the dominating dynamo eigenmodes,
the introduction of equatorial symmetry breaking considerably changes the
geometric character of the solutions. Around the dynamo threshold the leading
modes correspond to hemispherical dynamos even when the symmetry breaking is
small. This behavior can be explained by the approximate dipole-quadrupole
degeneration for the unperturbed problem.
  More complicated scenarios may occur in case of more realistic anisotropies
of \alpha- and \beta-effect or through non-linearities caused by the
back-reaction of the magnetic field (magnetic quenching).
