We carry out a systematic investigation of all the 2-loop integrals
occurring in the electron vertex in QED in the continuous $D$-dimensional
regularization scheme, for on-shell electrons, momentum transfer $t=-Q^2$
and finite squared electron mass $m_e^2=a$. We identify all the
Master Integrals (MI's) of the problem and write the differential equations
in $Q^2$ which they satisfy. The equations are expanded in powers of
$\epsilon = (4-D)/2$ and solved by the Euler's method of the variation
of the constants. As a result, we obtain the coefficients of the
Laurent expansion in $\epsilon$ of the MI's up to zeroth order
expressed in close analytic form in terms of Harmonic Polylogarithms