Equatorial Circular Motion of Charged Test Particles in a Weakly Magnetized Taub--NUT Background

We study circular motion of charged test particles on the equatorial slice of a Taub--NUT black hole with Manko--Ruiz parameter $C$, immersed in a weak external magnetic field introduced via Wald's prescription. Because the Taub--NUT metric is not reflection-symmetric about the equator once $l\neq 0$, generic charged orbits lie on cones $x=\cos\theta\neq 0$ rather than on the equatorial plane. We therefore analyse \emph{constrained} circular orbits obtained by imposing $x=\dot x=0$, and we exhibit in closed form the residual angular constraint that a fully self-consistent orbit would have to satisfy. Within this scope we derive the circularity and marginal-stability conditions and study how $B$ and $C$ shift the ISCO radius for prograde and retrograde branches. Increasing $B$ monotonically decreases $r_{\mathrm{ISCO}}$; the sign of the particle charge splits the two branches, with the ordering reversed between prograde and retrograde motion; and $C$ contributes only subleading corrections. The extension to self-consistent conical orbits is the natural direction for follow-up work.