Korteweg-de Vries equation

From [http://mathworld.wolfram.com/Korteweg-deVriesEquation.html]:

In fact, a transformation due to Gardner provides an algorithm for computing an infinite number of conserved densities of the KdV equation, which are connected to those of the so-called modified KdV equation through the Miura transformation v_x+v^2=u (Tabor 1989, p. 291). The Korteweg-de Vries equation also exhibits Galilean invariance.

See also [http://www.math.uwaterloo.ca/~karigiannis/papers/ist.pdf]:

In the course of attempting to solve the KdV equation exactly, it was discovered that the equation has an infinite sequence of nontrivial conservation laws, which we shall presently define.

from [http://en.wikipedia.org/wiki/Inverse_scattering_transform].

The so called turbulence closure problem has to do with the fact, that in fluid dynamics in principle an infinite number of conservation laws exist. Is there a connection? Can we write down a Korteweg-de Vries-like Equation, which generated the infinite conservation laws encountered in fluid dynamics?