  In this paper we examine a new class of five dimensional (5D) exact solutions
in extra dimension gravity possessing Lie algebroid symmetry. The constructions
provide a motivation for the theory of Clifford nonholonomic algebroids
elaborated in Ref. hep-th/0501217. Such Einstein-Dirac spacetimes are
parametrized by generic off--diagonal metrics and nonholonomic frames
(vielbeins) with associated nonlinear connection structure. They describe
self-consistent propagations of (3D) Dirac wave packets in 5D nonholonomically
deformed Taub NUT spacetimes and have two physically distinct properties: Fist,
the metrics are with polarizations of constants which may serve as indirect
signals for the presence of higher dimensions and/or nontrivial torsions and
nonholonomic gravitational configurations. Second, such Einstein-Dirac
solutions are characterized by new type of symmetries defined as
generalizations of the Lie algebra structure constants to nonholonomic Lie
algebroid and/or Clifford algebroid structure functions.
