  The universal hypermultiplet moduli space metric in the type-IIA superstring
theory compactified on a Calabi-Yau threefold is related to integrable systems.
The instanton corrections in four dimensions arise due to multiple wrapping of
BPS membranes and fivebranes around certain (supersymmetric) cycles of
Calabi-Yau. The exact (non-perturbative) metrics can be calculated in the
special cases of (i) the D-instantons (or the wrapped D2-branes) in the absence
of fivebranes, and (ii) the fivebrane instantons with vanishing charges, in the
absence of D-instantons. The solutions of the first type are governed by the
three-dimensional Toda equation, whereas the solutions of the second type are
governed by the particular Painleve VI equation.
