Riemann-Wirtinger integrals on the product of two one-dimensional complex tori

The Riemann-Wirtinger integral is an analogue of the hypergeometric integral defined on a one-dimensional complex torus. As a generalization, we define the Riemann-Wirtinger integral on the product of two one-dimensional complex tori. We study the structure of the twisted cohomology group associated with the Riemann-Wirtinger integral and derive a system of differential equations satisfied by this integral.