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Based on the generalized variational principle the system of the generalized Biot equations is derived for consistent account of fluid shear relaxation. Account of shear viscosity relaxation leads to existence of a couple shear propagation modes additionally to a couple of longitudinal modes as in the Biot approach. At this the one shear mode is an acoustical one, while the other shear mode has diffusive behavior at low frequencies. Phase velocity and attenuation factor for the second shear mode linearly depend on frequency in the low frequency limit that is different from analogous behavior of diffusive longitudinal mode with the root frequency behavior of analogous values.