  The Riemann-Hilbert approach to studying solutions of supergravity theories
allows us to associate spacetime independent monodromy matrices (matrices in
the Geroch group) with solutions that effectively only depend on two spacetime
coordinates. This offers insights into symmetries of supergravity theories, and
in the classification of their solutions. In this paper, we initiate a
systematic study of monodromy matrices for multi-center solutions of
five-dimensional U(1)$^3$ supergravity. We obtain monodromy matrices for a
class of collinear Bena-Warner bubbling geometries. We show that for this class
of solutions, monodromy matrices in the vector representation of SO(4,4) have
only simple poles with residues of rank two and nilpotency degree two. These
properties strongly suggest that an inverse scattering construction along the
lines of [arXiv:1311.7018 [hep-th]] can be given for this class of solutions,
though it is not attempted in this work. Along the way, we clarify a technical
point in the existing literature: we show that the so-called "spectral flow
transformations" of Bena, Bobev, and Warner are precisely a class of Harrison
transformations when restricted to the situation of two commuting Killing
symmetries in five-dimensions.
