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 funfbein, 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 (by higher dimension anholonomic gravitational interactions) 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.

We investigate higher rank Killing-Yano tensors showing that third rank Killing-Yano tensors are not always trivial objects being possible to construct irreducible Killing tensors from them. We give as an example the Kimura IIC metric were from two rank Killing-Yano tensors we obtain a reducible Killing tensor and from third rank Killing-Yano tensors we obtain three Killing tensors, one reducible and two irreducible. under some restrictions, pp-wave metrics and Siklos space-times admit non-generic supercharges [11]. On the other hand, there is a relation [12,13,14], called geometric duality, between spaces admitting irreducible Killing tensors of rank two and the spaces whose metrics are specified through those Killing tensors. Further generalizations of Killing tensors and their existence criteria were discussed [15,16,17,18]. Killing-Yano tensors and their corresponding Killing tensors have been studied extensively [19,20,21,22,23,24] in the related context of finding solutions of the Dirac-equation in non-trivial curved space-time. Moreover, in the context of generalized Dirac-type operators [25,26] the Killing-Yano tensors are indispensable tools.

In a Robertson -Walker space-time, a spinning particle model is investigated. It is shown that in a stationary case a class of new structures called f-symbols exists generating reducible Killing tensors and supersymmetry algebras.

In this paper we investigate a class of basic super-energy tensors, namely those constructed from Killing-Yano tensors, and give a generalization of super-energy tensors for cases when we start not with a single tensor, but with a pair of tensors.

We show that third rank Killing-Yano tensors are useful objects in the study of Diractype operators and their existence for a given background guarantees the absence of gravitational anomalies.