  We derive a non-perturbative, Lagrangian-level formulation of the double copy
in two spacetime dimensions. Our results elucidate the field theoretic
underpinnings of the double copy in a broad class of scalar theories which can
include masses and higher-dimension operators. An immediate corollary is the
amplitudes-level double copy at all orders in perturbation theory. Applied to
certain integrable models, the double copy defines an isomorphism between Lax
connections, Wilson lines, and infinite towers of conserved currents. We also
implement the double copy at the level of non-perturbative classical solutions,
both analytically and numerically, and present a generalization of the double
copy map that includes a fixed tower of higher-dimension corrections given by
the Moyal algebra.
