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Knot Manifolds of Double-Covariant Systems of Elliptic Equations and Preferred Orthonormal Three-Frames

We show that the problem of existence of preferred orthonormal frame in general relativity, which is formulated as a problem of solvability for the nonlinear system of elliptic equations, can be reduced to the linear problem. For the obtained system of equations and for more general one we find the necessary and sufficient conditions of existence and uniqueness of the solution for Dirichlet problem and the conditions of zeros absence for the solution, taking into account the availability of double symmetry. This allows in particular to prove existence of wide class of hypersurfaces on which the Sen–Witten equation and the Nester gauge are equivalent (up to the sign).