A Lax Operator for $d=2$ $N=2$ Supergravity

General relativity and supergravity become integrable systems after dimensional reduction to two spacetime dimensions. This means the equations of motion can be encoded in the flatness condition for a Lax operator. Nicolai and Warner found Lax operators for dimensionally reduced supergravity. They gave explicit formulas primarily for the case with $N=16$ supersymmetry in two dimensions (which corresponds to $N=8$ supergravity in four dimensions). In this note, we derive analogous results for the case with $N=2$ supersymmetry in two dimensions (which corresponds to $N=1$ supergravity in four dimensions). This is the simplest example of the general fact that supergravity becomes an integrable system after dimensional reduction to two dimensions.