  This paper is devoted to the symmetry and symmetry breaking properties of a
two-dimensional magnetic Schr{\"o}dinger operator involving an Aharonov-Bohm
magnetic vector potential. We investigate the symmetry properties of the
optimal potential for the corresponding magnetic Keller-Lieb-Thir-ring
inequality. We prove that this potential is radially symmetric if the intensity
of the magnetic field is below an explicit threshold, while symmetry is broken
above a second threshold corresponding to a higher magnetic field. The method
relies on the study of the magnetic kinetic energy of the wave function and
amounts to study the symmetry properties of the optimal functions in a magnetic
Hardy-Sobolev interpolation inequality. We give a quantified range of symmetry
by a non-perturbative method. To establish the symmetry breaking range, we
exploit the coupling of the phase and of the modulus and also obtain a
quantitative result.
