Non-BPS interactions from the type II one loop four graviton amplitude

We obtain T--duality invariant second order differential equations satisfied by the $D^8\mathcal{R}^4$ and $D^{10} \mathcal{R}^4$ interactions from the low energy expansion of the one loop four graviton amplitude in toroidally compactified type II string theory. The eigenvalues of these equations are completely determined by the structure of the one loop integrands. Unlike the BPS interactions, these non--BPS interactions satisfy Poisson equations having source terms that receive contributions from both the bulk and boundary of the worldsheet moduli space. We explicitly solve these equations in nine dimensions.