  We make use of the black hole holograph construction of [I. R\'acz,
Stationary black holes as holographs, Class. Quantum Grav. 31, 035006 (2014)]
to analyse the existence of Killing spinors in the domain of dependence of the
horizons of distorted black holes. In particular, we provide conditions on the
bifurcation sphere ensuring the existence of a Killing spinor. These conditions
can be understood as restrictions on the curvature of the bifurcation sphere
and ensure the existence of an axial Killing vector on the 2-surface. We obtain
the most general 2-dimensional metric on the bifurcation sphere for which these
curvature conditions are satisfied. Remarkably, these conditions are found to
be so restrictive that, in the considered particular case, the free data on the
bifurcation surface (determining a distorted black hole spacetime) is
completely determined by them. In addition, we formulate further conditions on
the bifurcation sphere ensuring that the Killing vector associated to the
Killing spinor is Hermitian. Once the existence of a Hermitian Killing vector
is guaranteed, one can use a characterisation of the Kerr spacetime due to Mars
to identify the particular subfamily of 2-metrics giving rise to a member of
the Kerr family in the black hole holograph construction. Our analysis sheds
light on the role of asymptotic flatness and curvature conditions on the
bifurcation sphere in the context of the problem of uniqueness of stationary
black holes. The Petrov type of the considered distorted black hole spacetimes
is also determined.
