By using anholonomic frames in (pseudo)-Riemannian spaces we define anisotropic extensions of Euclidean Taub-NUT spaces. With respect to coordinate frames such spaces are described by offdiagonal metrics which could be diagonalized by corresponding anholonomic transforms. We define the conditions when the 5D vacuum Einstein equations have as solutions anisotropic Taub-NUT spaces. The generalized Killing equations for the configuration space of anisotropically spinning particles (anisotropic spinning space) are analyzed. Simple solutions of the homogeneous part of these equations are expressed in terms of some anisotropically modified Killing-Yano tensors. The general results are applied to the case of the four-dimensional locally anisotropic Taub-NUT manifold with Euclidean signature. We emphasize that all constructions are for (pseudo)-Riemannian spaces defined by vacuum solutions, with generic anisotropy, of 5D Einstein equations, the solutions being generated by applying the moving frame method. 

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 the framework of the General Relativity we show that from three generalizations of Killing vector fields, namely f-symbols, symmetric Stäckel-Killing and antisymmetric Killing-Yano tensors, some conserved currents can be obtained through adequate contractions of the above mentioned objects with rank four tensors having the properties of Bel or Bel-Robinson tensors in Einstein spaces.

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.

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.