Circular motion of charged test particles in a weakly magnetized Taub--NUT background: From the equatorial slice to self-consistent conical orbits

We study circular motion of charged test particles around 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≠ 0, generic charged orbits lie on cones x=cosθ≠ 0 rather than on the equatorial plane. We first analyse constrained circular orbits obtained by imposing x= x=0, exhibit in closed form the residual angular constraint that such orbits violate, and derive the circularity and marginal-stability conditions on the imposed slice. We then remove the constraint: the radial and angular conditions are solved simultaneously for self-consistent circular orbits on cones x=x₀, and the true ISCO is obtained from the stability of these orbits. For $B=0$ the cone sits at x₀=2lE/ L with L=L-2lCE, and the constrained ISCO radius and energy coincide exactly with the self-consistent ones; for B≠0 the difference grows as B² and stays below 10⁻³M over the parameter range studied. The Manko--Ruiz parameter is pure gauge at $B=0$, where it only relabels the angular momentum, and enters physical quantities at B≠0 only through the electrostatic part Aₜ∝ lCB of the Wald potential. Increasing B monotonically decreases rISCO; the sign of eBLeff splits the branches, with anti-Larmor orbits reaching closer to the horizon; and C contributes only at order $eBl$.