  We consider the perturbative contributions to the R^4, D^4 R^4 and D^6 R^4
interactions in toroidally compactified type II string theory. These BPS
interactions do not receive perturbative contributions beyond genus three. We
derive Poisson equations satisfied by these moduli dependent string amplitudes.
These T--duality invariant equations have eigenvalues that are completely
determined by the structure of the integrands of the multi--loop amplitudes.
The source terms are given by boundary terms of the moduli space of Riemann
surfaces corresponding to both separating and non--separating nodes. These are
determined directly from the string amplitudes, as well as from the logarithmic
divergences of maximal supergravity. We explicitly solve these Poisson
equations in nine and eight dimensions.
