A gauged bi-differential calculus over an associative (and not necessarily
commutative) algebra A is an N-0-graded left A-module with two covariant de
rivatives acting on it which, as a consequence of certain (e.g. nonlinear d
ifferential) equations, are flat and anticommute. As a consequence, there i
s an iterative construction of generalized conserved currents. We associate
a gauged bi-differential calculus with the Korteweg-de Vries equation and
use it to compute conserved densities of this equation.