  We construct and study singular functions in strong-field $QED$ with two
external electromagnetic fields that represent principally different types of
external backgrounds, the first one belongs to the class of so-called
$t$-potential electric steps (electric-like fields that are switched on and off
at initial and final time instants), and the second one belongs to the class of
so-called $x$-potential electric steps (time-independent electric-like fields
of constant direction that are concentrated in a restricted spatial area). As
the first background ($T$-constant electric field) is chosen an uniform
electric field which acts during a finite time interval $T$ , whereas as the
second background ($L$-constant electric field) is chosen a constant electric
field confined between two capacitor plates separated by a large distance $L$.
For the both cases we find \textrm{in}- and \textrm{out}-solutions of the Dirac
equation in terms of light cone variables. With the help of these solutions, we
construct Fock-Schwinger proper-time integral representations for all the
singular functions that provide nonperturbative (with respect to the external
backgrounds) calculations of any transition amplitudes and mean values of any
physical quantities. Considering calculations in the $T$-constant field and in
the $L$-constant field as different regularizations of the corresponding
calculations in the constant uniform electric field, we have demonstrated their
equivalence for sufficiently large $T$ and $L$.
