Belinski and Zakharov (1978) -- Inverse Scattering Method -- Soliton solutions on arbit. backgr. -- Riemann – Hilbert problem + linear singular integral equations Integrability - ? -- R.Geroch –conjecture (1972) -- W.Kinnersley –inf.dim. algebra (1977…) -- D.Maison - Lax pair +conjecture (1978) Vacuum Symmetries: Later results: -- Backlund transformations (Harrison 1978, Neugebauer 1979) -- Homogeneous Hilbert problem (Hauser & Ernst, 1979+Sibgatullin 1984) -- Monodromy Transform + linera singular integral equations (GA 1985) -- Finite-gap solutions (Korotkin&Matveev 1987, Neugebauer&Meinel 1993) -- Charateristic initial value problem (GA & Griffiths 2001) --- Non-vacuum integrable reductions of Einstein’s field equations

G. Alekseev

Coordinates:

Analytical structure of on w – plane: GA, Sov. Phys (1985) ; 1)

15 Analytical structure of on the spectral plane

Monodromy data of a given solution Monodromy data for solutions:

17 GA, Sov.Phys.Dokl. 1985;Proc. Steklov Inst. Math. 1988; Theor.Math.Phys )

18 Generic data:Analytically matched data: Unknowns: Rational, analytically matched data:

GA and V.Belinski Phys.Rev. D (2007) 1) In equilibrium:

21 Free space of the monodromy data Space of solutions For any holomorphic local solution near, Theorem 1. Is holomorphic on and the ``jumps’’ of on the cuts satisfy the H lder condition and are integrable near the endpoints. posess the same properties GA, Sov.Phys.Dokl. 1985;Proc. Steklov Inst. Math. 1988; Theor.Math.Phys )

22 *) For any holomorphic local solution near, Theorem 2. possess the local structures Fragments of these structures satisfy in the algebraic constraints and and the relations in boxes give rise later to the linear singular integral equations. (for simplicity we put here ) where are holomorphic on respectively. In the case N-2d we do not consider the spinor field and put *)

23 Theorem 3. For any local solution of the ``null curvature'' equations with the above Jordan conditions, the fragments of the local structures of and on the cuts should satisfy where the dot for N=2d means a matrix product and the scalar kernels (N=2,3) or dxd-matrix (N=2d) kernels and coefficients are where and each of the parameters and runs over the contour ; e.g.: In the case N-2d we do not consider the spinor field and put *)

24 Theorem 4. For arbitrarily chosen extended monodromy data – the scalar functions and two pairs of vector (N=2,3) or only two pairs of dx2d and 2dxd – matrix (N=2d) functions and holomorphic respectively in some neighbor-- hoods and of the points and on the spectral plane, there exists some neighborhood of the initial point such that the solutions and of the integral equations given in Theorem 3 exist and are unique in and respectively. The matrix functions and are defined as is a normalized fundamental solution of the associated linear system with the Jordan conditions.