  Two different constructions generating the low-energy expansion of genus-one
configuration-space integrals appearing in one-loop open-string amplitudes have
been put forward in \rcites{Mafra:2019xms, *Mafra:2019ddf, Broedel:2019gba}. We
are going to show that both approaches can be traced back to an elliptic system
of Knizhnik--Zamolodchikov--Bernard(KZB) type on the twice-punctured torus. We
derive an explicit all-multiplicity representation of the elliptic KZB system
for a vector of iterated integrals with an extra marked point and explore
compatibility conditions for the two sets of algebra generators appearing in
the two differential equations.
