  We obtain a second order differential equation on moduli space satisfied by
certain modular graph functions at genus two, each of which has two links. This
eigenvalue equation is obtained by analyzing the variations of these graphs
under the variation of the Beltrami differentials. This equation involves seven
distinct graphs, three of which appear in the integrand of the
$D^8\mathcal{R}^4$ term in the low momentum expansion of the four graviton
amplitude at genus two in type II string theory.
