  In view of a result recently published in the context of deformation theory
of linear Hamiltonian systems, we reconsider the eigenvalue problem associated
to the angular equation arising after the separation of the Dirac equation in
the Kerr metric and we show how efficiently a quasi-linear first order PDE for
the angular eigenvalues can be derived. We also prove that it is not possible
to obtain an ordinary differential equation for the eigenvalues where the role
of the independent variable is played by the particle energy or the black hole
mass. Finally, we construct new perturbative expansions for the eigenvalues in
the Kerr case and obtain an asymptotic formula for the eigenvalues in the case
of a Kerr naked singularity.
