  We consider the Riemann-Hilbert factorization approach to the construction of
Weyl metrics in four space-time dimensions. We present, for the first time, a
rigorous proof of the remarkable fact that the canonical Wiener-Hopf
factorization of a matrix obtained from a general (possibly unbounded)
monodromy matrix, with respect to an appropriately chosen contour, yields a
solution to the non-linear gravitational field equations. This holds regardless
of whether the dimensionally reduced metric in two dimensions has Minkowski or
Euclidean signature. We show moreover that, by taking advantage of a certain
degree of freedom in the choice of the contour, the same monodromy matrix
generally yields various distinct solutions to the field equations. Our proof,
which fills various gaps in the existing literature, is based on the solution
of a second Riemann-Hilbert problem and highlights the deep role of the
spectral curve, the normalization condition in the factorization and the choice
of the contour. This approach allows us to construct explicit solutions,
including new ones, to the non-linear gravitational field equations, using
simple complex analytic results.
