We have derived the far-field asymptotic formula for the S-wave elastodynamic Green function in the kiss singularity in anisotropic media. In contrast to standard asymptotics in regular directions the derived formula is more complex and expressed in the form of 1-D integral. This integral is specified for the kiss singularity along the symmetry axis in transversely isotropic media and along the fourfold symmetry axes in the tetragonal and cubic media. The shape of the slowness surface in the singularity is regular in transverse isotropy and the amplitude of the Green function is expressed by means of the Gaussian curvature of this surface in the singularity. However, the shape of the slowness surface of S waves is irregular in tetragonal or cubic media. In this case the Gaussian curvature is not defined in the singularity. Therefore, a generalized Gaussian curvature is introduced and the amplitude of the Green function is expressed by means of this new quantity. For regular directions, both the generalized and standard Gaussian curvatures yield the same value.
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