We prove that nonassociative star deformed vacuum Einstein equations can be decoupled and integrated in certain general forms on phase spaces involving real R‐flux terms induced as parametric corrections on base Lorentz manifold spacetimes. The geometric constructions are elaborated with parametric (on respective Planck, ℏ, and string, , constants) and nonholonomic dyadic decompositions of fundamental geometric and physical objects. This is our second partner work on elaborating nonassociative geometric and gravity theories with symmetric and nonsymmetric metrics, (non) linear connections, star deformations defined by generalized Moyal‐Weyl products, endowed with quasi‐Hopf algebra, or other type algebraic and geometric structures, and all adapted to nonholonomic distributions and frames. We construct exact and parametric solutions for nonassociative vacuum configurations (with nontrivial or effective cosmological constants) defined by star deformed generic off‐diagonal (non) symmetric metrics and (generalized) nonlinear and linear connections. The coefficients of geometric objects defining such solutions are determined by respective classes of generating and integration functions and constants and may depend on all phase space coordinates [spacetime ones, ; and momentum like variables, )]. Quasi‐stationary configurations are stated by solutions with spacetime Killing symmetry on a time like vector but with possible dependencies on momentum like coordinates on star deformed phase spaces. This geometric techniques of decoupling and integrating nonlinear systems of physically important partial differential equations (the anholonomic frame and connection deformation method, AFCDM) is applied in our partner works for constructing nonassociative and locally anisotropic generalizations of black hole and cosmological solutions and elaborating geometric flow evolution and classical and quantum information theories.

中文翻译：

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具有星型和R轴通量参数变形的非缔合真空相空间引力方程的解耦和可积性

我们证明了非缔合星形变形真空爱因斯坦方程可以在特定的一般形式上解耦和积分，包括作为基本Lorentz流形时空的参数校正而引起的实际R磁通项的相空间。用参数（分别在Planck，ℏ和弦上，，常数）和基本几何和物理对象的非完整二元分解。这是我们的第二个合作伙伴研究，目的是通过对称和非对称度量，（非）线性连接，由广义Moyal-Weyl乘积定义的恒星形变，赋予拟霍普夫代数或其他类型的代数和几何结构，阐述非关联几何和重力理论，并且都适应了非完整的分布和框架。我们为星形变形通用非对角（非）对称度量和（广义）非线性和线性连接定义的非关联真空配置（具有非平凡或有效的宇宙学常数）构造精确和参数解。; 和动量之类的变量，）]。准平稳配置由具有时空Killing对称性的解表示，该解对像矢量一样的时间，但可能依赖于动量，如星形变形相空间上的坐标。这种将物理上重要的偏微分方程的非线性系统解耦和积分的几何技术（完整框架和连接变形方法，AFCDM）被用于我们的合作伙伴工作中，以构造黑洞和宇宙学解的非缔合和局部各向异性概化，并阐述几何流动演化以及经典和量子信息理论。