Bi-differential calculus and the kdv equation

A gauged bi-differential calculus over an associative (and not necessarily commutative) algebra A is an No-graded left A-module with two covariant derivatives acting on it which, as a consequence of certain (e.g. nonlinear differential) equations, are flat and anticommute. As a consequence, there is 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.