Spherical photon orbits around the Kerr-like black hole in Einstein-Bumblebee gravity

In this paper, we investigate the photon orbits around a Kerr-like black hole in Einstein-Bumblebee gravity, where Lorentz symmetry is spontaneously broken. By solving the Hamilton-Jacobi equation, we derive a sixth-order polynomial that governs the photon motion, explicitly dependent on the rotation parameter $u$, the Lorentz violation parameter $\ell$, and the effective inclination angle $v$. We analyze photon orbit configurations in polar, equatorial, and general inclined planes, identifying significant deviations from the Kerr solution. In the polar and equatorial planes, we identify distinct photon orbit configurations and analyze their dependence on model parameters. For general inclined orbits, we find a critical inclination angle $v$ that determines the number and location of photon orbits in both extremal and non-extremal cases. All photon orbits are radially unstable, and the critical impact parameter decreases with increasing Lorentz violation, potentially providing observable signatures to differentiate Einstein-Bumblebee gravity from general relativity.