Based on symmetries Taub-NUT shares with Bertrand spacetime, we cast it as the latter with magnetic fields. Its nature as a Bianchi-IX gravitational instanton and other related geometrical properties are reviewed. We provide an easy derivation and comparison between the spatial Killing-Yano tensors deduced from first-integrals and the corresponding hyperkähler structures and finally verify the existence of a graded Lie-algebra structure via Schouten-Nijenhuis brackets. The arguments of the energy of these four patterns are obtained based on the material symmetry group of the considered structures.
"Taub-NUT as Bertrand Spacetime with Magnetic Fields." J. Geom. Symmetry Phys. 41 33 - 67, 2016. https://doi.org/10.7546/jgsp-41-2016-33-67