New Lorentzian Taub-NUT and Euclidean Eguchi-Hanson Solutions in $f(R)$ gravity

A New Taub-NUT black hole in multiple coordinate systems is solved and analyzed. This black-hole is found to be an extension of the Clifton-Barrow metric for the specific case that $\delta =-1\mathrm{/}2$. We look at physical properties such as test particle orbits, thermodynamics, and various space-time limits. In addition, we show an Eguchi-Hanson type-II $f(R)$ solution related to the found Taub-NUT framework.