Penrose transform and general wave equations In the late 1960's Penrose developed twistor theory, which (amongst other things) led to an exceptional description for solutions to the wave equation on Minkowski space via the so-called Penrose transform;
If 
\begin{equation}
u(x,y,z,t) = \frac {1} {2 \pi i} \oint_{\Gamma \subset \mathbb{C} \mathbb{P}^{1}} f(-(x+iy) + \lambda (t-z), (t+z) + \lambda (-x + i y), \lambda ) d \lambda, \,\,\,\,\,\,\,\,\,\, (1)
\end{equation}
where $\Gamma \subset \mathbb{C} \mathbb{P}^{1}$ is a closed contour and $f$ is holomorphic on $\mathbb{C} \mathbb{P}^{1}$ except at some number of poles, then $u$ satisfies the Minkowski wave (Laplace-Beltrami) equation $\square_{\eta} u = 0$.
I am aware that there are a number of works in the literature describing twistor theory on curved manifolds, but have not seen explicit constructions along the lines of (1) such that the function $u$ satisfies a wave equation of the form $\square_{g} u = 0$ for (Lorentzian) metric $\boldsymbol{g}$. 
Is it known how to explicitly construct contour integrals similar to $(1)$ for some class of metrics $\boldsymbol{g}$? What about when $\boldsymbol{g}$ is Einstein (e.g. Schwarzschild), in particular? Are there topological obstructions in spacetimes $I \times \Sigma$? What about de-Sitter space?
 A: I believe you can express the Penrose transform using a contour integral by
expressing Serre duality in terms of Dolbeault cohomology. For example, IIRC,
on a half-conformally-flat
4-manifold $M$ with twistor space $Z$, functions solving the Laplacian equation correspond to elements
of the sheaf cohomology group $H^1(Z, \mathcal{O}(-2))$ and the evaluation at a point
$x \in M$, with corresponding twistor line $P_x \subset Z$, is to restrict to:
$$
  H^1(P_x, \mathcal{O}(-2)) \simeq H^0(P_x, \mathcal{O})^* \simeq \mathbb{C}.
$$
If we represent our element of $H^1(P_x, \mathcal{O}(-2))$ by a form in
$H^{0,1}(P_x, \mathcal{O}(-2))$,
then choosing a contour that does not meet the singularities from the section $\mathcal{O}(-2)$, the Serre duality map above yields an contour integral by Stokes's theorem.
A: A construction similar to but not exactly the same as the one you describe is for instance the one in 
Hitchin, Karlhede, Lindstrom and Rocek - Hyperkähler metrics and supersymmetry, Commun. Math. Phys. 108, 535-589 (1987)
see equation (2.2). The metrics constructed there are hyperkahler metrics with toric symmetries of rank equal to their quaternionic dimension. The contour integral satisfies the Laplace and not the wave equation, and the resulting metrics have in general signature $(4n,4m)$.
