  Let f,g be two algebraically independent regular functions from the smooth
affine complex variety U to the affine line. The associated exponential
Gauss-Manin systems on the affine line are defined to be the cohomology sheaves
of the direct image of the exponential differential system $\mathcal{O}_U e^g $
with respect to f. We prove that its holomorphic solutions admit
representations in terms of period integrals over topological chains with
possibly closed support and with rapid decay condition.
