  We evaluate the master integrals for the two-loop, planar box-diagrams
contributing to the elastic scattering of muons and electrons at
next-to-next-to leading-order in QED. We adopt the method of differential
equations and the Magnus exponential series to determine a canonical set of
integrals, finally expressed as a Taylor series around four space-time
dimensions, with coefficients written as combination of generalised
polylogarithms. The electron is treated as massless, while we retain full
dependence on the muon mass. The considered integrals are also relevant for
crossing-related processes, such as di-muon production at $e^+ e^-$-colliders,
as well as for the QCD corrections to $top$-pair production at hadron
colliders.
