Gravitational and electromagnetic solitonsMonodromy transform approachSolution of the characteristic initial value problem;Colliding gravitational and electromagnetic waves Many languages of integrabilitySolutions for black holes in the external fields

mathematical context: - infinite hierarchies of exact solutions, - initial and boundary value problems, - asymptotical behaviourIntegrable cases: - Vacuum gravitational fields - Einstein Maxwell - Weyl fields - Ideal fluid with - some string gravity models physical context: - supeposition of stat. axisymm. fields, - nonlinear interacting waves, - inhomogeneous cosmological models

Associated linear systems and ``spectral problems Infinite-dimensional algebra of internal symmetries Solution generating procedures (arbitrary seed): -- Solitons, -- Backlund transformations, -- Symmetry transformations Infinite hierarchies of exact solutions -- Meromorfic on the Riemann sphere -- Meromorfic on the Riemann surfaces (finite gap solutions) Prolongation structures Geroch conjecture Riemann Hielbert and Homogeneous Hilbert problems, Various linear singular integral equation methods Initial and boundary value problems -- Characteristic initial value problems -- Boundary value problems for stationary axisymmetric fields Twistor theory of the Ernst equation

SU(2,1) symmetric form of dynamical equations Einstein Maxwell fields: the Ernst-like equations1)W.Kinnersley, J. Math.Phys. (1973) 1)

1)Isometry group with 2-surface orthogonal orbits:The Einsteins field equations:-- the constraint equations -- the dynamical equations-- the dynamical equations

Geometrically defined coordinates:Generalized Weyl coordinates:

Belinski Zakharov vacuum solitonsEinstein Maxwell solitonsExamples of soliton solutionsIntegrable reductions of Einstein equations

Belinski Zakharov form of reduced vacuum equationsKinnersley self-dual form of the reduced vacuum equations2x2-matrix form of self-dual reduced vacuum equationsErnst vacuum equation

Associated spectral problem V.Belinski & V.Zakharov,, JETP 1978; 1979 ; 1)1)Dynamical equations for vacuum Dressing method for constructing solutions

Riemann problem for dressing matrixLinear singular integral equationsConstraints for dressing matrix:V.Belinski & V.Zakharov,, JETP 1978; 1979 ; 1)Formulation of the matrix Riemann problem1)

V.Belinski & V.Zakharov,, JETP 1978; 1979 ; 1)( - solitons)Vacuum solitons1)Soliton ansatz for dressing matrix

GA, Sov.Phys.Dokl. (1981) ; 1)1)Stationary axisymmetric solitons on the Minkowski background:a set of 4 N arbitrary real or pairwise complex conjugated constants

Integrable reductions of Einstein-Maxwell equationsSpacetime metric and electromagnetic potential:

Ernst potentials :Ernst equations:

3x3-matrix form of Einstein Maxwell equaations

1)GA, JETP Lett.. (1980); Proc. Steklov Inst. Math. (1988); Physica D. (1999)1)For vacuum:

(w - solitons)Soliton ansatz for dressing matrixGA, JETP Lett. (1980); Proc. Steklov Inst. Math. (1988); Physica D. (1999) 1)1)Dressing matrix :--- a set of 3 N arbitrary complex constants

-- Superextreme part of the Kerr-Newman solution-- Interaction of two superextreme Kerr-Newman sources-- mass -- NUT-parameter -- angular momentum-- electric charge-- magnetic chargeGA, Proc. Steklov Inst. Math. (1988); Physica D. (1999) 1)1)

-- Interaction of two superextreme Kerr-Newman sources