Integrating simple genus two string invariants over moduli space

We consider an Sp(4,Z) invariant expression involving two factors of the Kawazumi--Zhang (KZ) each which is a modular graph with one link, and four derivatives on moduli space genus Riemann surfaces. Manipulating it, we show that integral over linear combination links square KZ reduces to boundary integral. also three six space, from deduce In both cases, term completely determined by invariant. integrals vanish.