  We apply a new general method of anholonomic frames with associated nonlinear
connection structure to construct new classes of exact solutions of
Einstein-Dirac equations in five dimensional (5D)gravity. Such solutions are
parametrized by off-diagonal metrics in coordinate (holonomic) bases, or,
equivalently, by diagonal metrics given with respect to some anholonomic frames
(pentads, or funfbiends, satisfing corresponding constraint relations). We
consider two possibilities of generalization of the Taub NUT metric in order to
obtain vacuum solutions of 5D Einsitein equations with effective
renormalization of constants having distinguished anisotropies on an angular
parameter or on extra dimension coordinate. The constructions are extended to
solutions describing self-consistent propagations of 3D Dirac wave packets in
5D anisotropic Taub NUT spacetimes. We show that such anisotropic
configurations of spinor matter can induce gravitational 3D solitons being
solutions of Kadomtsev-Petviashvili or of sine-Gordon equations.
