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Revision with unchanged content. This work is devoted to the field of turbulence modeling. It is focused on the development of the complete second-order turbulence closure model. The alternative approach for obtaining the linear anisotropy invariant map based on the eigenvalues of the second-order tensors, which represent the important turbulent quantities like the Reynolds stress, turbulent dissipation tensor and pressure-strain correlation term, is proposed. The general modeling relationships between the corresponding tensors are derived using the kinematic constraints and validated for the particular classes of turbulent flows. The new modeling framework, based on application of the linear anisotropy invariant considerations, is placed in the context of the most recent Reynolds stress transport models, so that the innovation and advantages of the proposed methodology can be evaluated directly.