It is known that in integrable systems existence of a Lax pair $$ \dot{L} = [M,L]$$ leads to the existence of conserved quantities $$ \frac{d}{dt}\mathrm{Tr}(L^k) = \sum \mathrm{Tr}(L^i [M,L] L^{k-i})$$ Suppose that I have a Lax pair with differential operator, for example, KdV $$ L = -\frac{d^2}{dx^2}+u(x,t), \quad A = 4\frac{d^3}{dx^3}-3(u\frac{d}{dx}+\frac{d}{dx}u)$$
Is it possible to apply the same way of constructing conserved quantities in this case? If it is so, what is trace?
I tried considering that it shall be something similar to quantum mechanics: $\mathrm{Tr} A = \int dx\langle x| A |x\rangle$, where $\langle x' |x\rangle = \delta(x-x'),$ but didn't succeed in calculating this integral.