A non-trivial conservation law with a trivial characteristic

As far as I am aware, no nontrivial conservation laws surviving to the second page of Vinogradov's C-spectral sequence have been established. It turns out that presymplectic structures that cannot be described in terms of cosymmetries produce such conservation laws for closely related overdetermined systems. In particular, the presymplectic structure $D_x$ of the potential mKdV equation gives rise to such a conservation law for the overdetermined system $u_t = 4u_x^3 + u_{xxx}$, $u_y = 0$. While this example is somewhat degenerate, it may be one of the simplest systems exhibiting this phenomenon.