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Geometric methods for constructing exact solutions of motion equations withfirst order $\alpha ^{\prime}$ corrections to the heterotic supergravity actionimplying a non-trivial Yang-Mills sector and six dimensional, 6-d,almost-K\"ahler internal spaces are studied. In 10-d spacetimes, generalparametrizations for generic off-diagonal metrics, nonlinear and linearconnections and matter sources, when the equations of motion decouple in verygeneral forms are considered. This allows us to construct a variety of exactsolutions when the coefficients of fundamental geometric/physical objectsdepend on all higher dimensional spacetime coordinates via correspondingclasses of generating and integration functions, generalized effective sourcesand integration constants. Such generalized solutions are determined by genericoff-diagonal metrics and nonlinear and/or linear connections. In particular, asconfigurations which are warped/compactified to lower dimensions and forLevi-Civita connections. The corresponding metrics can have (non) Killingand/or Lie algebra symmetries and/or describe (1+2)-d and/or (1+3)-d domainwall configurations, with possible warping nearly almost-K\"ahler manifolds,with gravitational and gauge instantons for nonlinear vacuum configurations andeffective polarizations of cosmological and interaction constants encodingstring gravity effects. A series of examples of exact solutions describinggeneric off-diagonal supergravity modifications to black hole/ ellipsoid andsolitonic configurations are provided and analyzed. We prove that it ispossible to reproduce the Kerr and other type black solutions in generalrelativity (with certain types of string corrections) in 4-d and to generalizethe solutions to non-vacuum configurations in (super) gravity/ string theories.

Off-Diagonal Deformations of Kerr Metrics and Black Ellipsoids in Heterotic Supergravity