$J^\infty=(x,v,v_x,\ldots)$.

The compatible Hamiltonian operators – yield the hereditary recursion operator $$\label{mkdv-Rop}
\R=\E\H\inv=D_x^2+v^2 +v_xD\inv_x v .$$ This operator and both of the Hamiltonian operators are invariant under $x$-translations applied to $v$, which corresponds to the invariance of the mKdV equation under the symmetry operator $\X=-v_x\partial/\partial_v$ representing infinitesimal $x$-translations. The recursion operator can be combined with this symmetry to express the mKdV equation as a flow $$v_t=\R({-v_x})=-\tfrac{3}{2}v^2 v_x -v_{xxx} .$$ Higher flows are generated by $$\label{mkdv-hierarchy}
P^{(n)}=\R^n({-v_x}),
\quad
n=1,2,\ldots$$ corresponding to an integrable hierarchy of equations $v_t= P^{(n)}$, where the $n=1$ flow is the mKdV equation and each successive flow $n\geq 2$ inherits a bi-Hamiltonian structure $P^{(n)} = \H(\delta H^{(n)}/\delta v)=\E(\delta H^{(n-1)}/\delta v)$ which comes from Magri’s theorem, with $H^{(0)}=E$ and $H^{(1)}=H$. The gradients of these Hamiltonians are generated by $\delta H^{(n)}/\delta v = {\R^*}^n(v) = Q^{(n)}$, $n=1,2,\ldots$, in terms of the adjoint recursion operator $$\label{mkdv-adRop}
\R^*=\H\inv\E=D_x^2+v^2 -vD\inv_x v_x$$ where $Q^{(0)}=\delta H^{(0)}/\delta v=\delta E/\delta v = v$.

It is useful to observe that $P^{(0)}=-v_x$ also can be expressed as a bi-Hamiltonian flow, in a certain formal sense. Its first Hamiltonian structure is simply $P^{(0)} = \H(\delta H^{(0)}/\delta v)$, while its second Hamiltonian structure arises from the relation $\E(0) = -v_xD\inv_x(0)$ if 