We propose a new setup for creating Majorana bound states in a
two-dimensional electron gas Josephson junction. Our proposal relies
exclusively on a supercurrent parallel to the junction as a mechanism of
breaking time-reversal symmetry. We show that combined with spin-orbit
coupling, supercurrents induce a Zeeman-like spin splitting. Further, we
identify a new conserved quantity—charge-momentum parity—that prevents
the opening of the topological gap by the supercurrent in a straight
Josephson junction. We propose breaking this conservation law by adding
a third superconductor, introducing a periodic potential, or making the
junction zigzag-shaped. By comparing the topological phase diagrams and
practical limitations of these systems we identify the zigzag-shaped
junction as the most promising option.