  We examine the motion of the massless scalar field and nearly bound null
geodesics in the near-ring region of a black hole, which may possess either
acceleration or a gravitomagnetic mass. Around such black holes, the photon
ring deviates from the equatorial plane. In the large angular momentum limit,
we demonstrate that the massless scalar field exhibits an emergent conformal
symmetry in this near-ring region. Additionally, in the nearly bound limit, we
observe the emergence of a conformal symmetry for the null geodesics that
constitute the photon ring in the black hole image. These findings suggest that
the hidden conformal symmetry, associated with the Lie algebra $\mathfrak{s
l}(2, \mathbb{R})$, persists even for black holes lacking north-south
reflection symmetry, thereby broadening the foundation of photon ring
holography. Finally, we show that the conformal symmetry also emerges for
nearly bound timelike geodesics and scalar fields in proximity to the particle
ring, and with specific mass around a Schwarzschild black hole.
