  Half of the Bondi-Metzner-Sachs (BMS) transformations consist of
orientation-preserving conformal homeomorphisms of the extended complex plane
known as fractional linear (or Mobius) transformations. These can be of 4
kinds, i.e. they are classified as being parabolic, or hyperbolic, or elliptic,
or loxodromic, depending on the number of fixed points and on the value of the
trace of the associated 2x2 matrix in the projective version of the SL(2,C)
group. The resulting particular forms of SL(2,C) matrices affect also the other
half of BMS transformations, and are used here to propose 4 realizations of the
asymptotic symmetry group that we call, again, parabolic, or hyperbolic, or
elliptic, or loxodromic. In the second part of the paper, we prove that a
subset of hyperbolic and loxodromic transformations, those having trace that
approaches infinity, correspond to the fulfillment of limit-point condition for
singular Sturm-Liouville problems. Thus, a profound link may exist between the
language for describing asymptotically flat space-times and the world of
complex analysis and self-adjoint problems in ordinary quantum mechanics.
