Hamilton–Jacobi separability of the unequal-factor physical metric

Determine whether the Hamilton–Jacobi equation for geodesic motion in the reconstructed rotating Palatini Kalb–Ramond physical metric is separable when the deformation factors A=1+\xi_1 b^2 and C=A+\xi_2 b^2/2 are unequal, and whether an additional separation constant exists in that case.

Background

The paper reconstructs a physical metric from a Kerr auxiliary metric and a closed, corotating Kalb–Ramond two-form. When the deformation factors are unequal, the reconstructed geometry is generally not Kerr and contains mixed metric components that couple radial, polar, and azimuthal motion. Consequently, the authors formulate geodesic dynamics using the full Hamiltonian rather than assuming Kerr-like separated equations or an additional Carter-type constant.

The paper explicitly states that separability has not been established for this unequal-factor geometry. The subsequent geodesic analysis therefore treats the radial and polar dynamics as coupled and obtains numerical photon trajectories and critical curves directly from the full Hamiltonian. Establishing separability, or proving its failure and identifying the resulting integrability structure, remains unresolved.