In this paper the final stage of the Petrov classification is carried out. As it is known, the Killing vector fields specify infinitesimal transformations of the group of motions of space $V_4$. In the case when in the homogeneous space $V_4$ the group of motions $G_3$ acts simply transitive, the geometry of the non-isotropic hypersurface is determined by the geometry of the transitivity space $V_3$ of the group $G_3$. In this case, the metric tensor of the space $V_3$ can be given by a nonholonomic reper consisting of three independent vectors $\ell_{ (a)}^\alpha$, which define the generators of the group $G_3$ of finite transformations in the space $V_3$. The representation of the metric tensor of $V_4$ spaces by means of vector fields $\ell_{ (a)}^\alpha$ has a great physical meaning and allows to simplify substantially the equations of mathematical physics in such spaces. Therefore, the Petrov classification should be complemented by the classification of vector fields $\ell_{ (a)}^\alpha$ connected to Killing vector fields. For homogeneous spaces this problem has been largely solved. A complete solution of this problem is presented in the present paper, where the Petrov classification for homogeneous spaces in which the group $G_3$, which belongs to type $VIII$ according to the Petrov classification, acts simply transitively, is refined. In addition, the complete classification of vector fields $\ell_{ (a)}^\alpha$ for spaces $V_4$ in which the group $G_3$ acts simply transitivity on isotropic hypersurfaces.}
Categories: gr-qc