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We present new classes of exact solutions with noncommutative symmetriesconstructed in vacuum Einstein gravity (in general, with nonzero cosmologicalconstant), five dimensional (5D) gravity and (anti) de Sitter gauge gravity.Such solutions are generated by anholonomic frame transforms and parametrizedby generic off-diagonal metrics. For certain particular cases, the new classesof metrics have explicit limits with Killing symmetries but, in general, theymay be characterized by certain anholonomic noncommutative matrix geometries.We argue that different classes of noncommutative symmetries can be induced byexact solutions of the field equations in 'commutative' gravity modeled by acorresponding moving real and complex frame geometry. We analyze two classes ofblack ellipsoid solutions (in the vacuum case and with cosmological constant)in 4D gravity and construct the analytic extensions of metrics for certainclasses of associated frames with complex valued coefficients. The third classof solutions describes 5D wormholes which can be extended to complex metrics incomplex gravity models defined by noncommutative geometric structures. Theanholonomic noncommutative symmetries of such objects are analyzed. We alsopresent a descriptive account how the Einstein gravity can be related to gaugemodels of gravity and their noncommutative extensions and discuss suchconstructions in relation to the Seiberg-Witten map for the gauge gravity.Finally, we consider a formalism of vielbeins deformations subjected tononcommutative symmetries in order to generate solutions for noncommutativegravity models with Moyal (star) product.Comment: latex2e, 53 pages, the variant to be published in J. Math. Phy

Exact solutions with noncommutative symmetries in Einstein and gauge gravity