  We apply the method of isomonodromy to study the scattering of a generic
Kerr-NUT-(A)dS black hole. For generic values of the charges, the problem is
related to the connection problem of the Painlev\'e VI transcendent. We review
a few facts about Painlev\'e VI, Garnier systems and the Hamiltonian structure
of flat connections in the Riemann sphere. We then outline a method for
computing the scattering amplitudes based on Hamilton-Jacobi structure of
Painlev\'e, and discuss the implications of the generic result to black hole
complementarity.
