$D\inv_x(0)$ is redefined by the addition of some non-zero constant, so that $D\inv_x(0)=c\neq 0$. This yields $P^{(0)} = c^{-1}\E(0)$, where the corresponding Hamiltonian is trivial. Then this flow has the bi-Hamiltonian structure $$\label{mkdv-rootflow}
P^{(0)} = \H(\delta H^{(0)}/\delta v) =c^{-1}\E(0) .$$

Each higher flow in the mKdV hierarchy corresponds to a higher symmetry operator $\X^{(n)}=P^{(n)}\partial/\partial_v$, $n=1,2,\ldots$, all of which represent infinitesimal symmetries that are admitted by every equation $v_t= P^{(n)}$, $n=1,2,\ldots$, in the hierarchy. Each Hamiltonian $H^{(n)}$, $n=0,1,2,\ldots$, of the flows in the hierarchy corresponds to a conservation law $\frac{d}{dt} H^{(n)}=0$, all of which hold for every equation $v_t= P^{(n)}$, $n=1,2,\ldots$, in the hierarchy.

The mKdV equation has the scaling symmetry $v\to\lambda v$, $x\to\lambda^{-1}x$, $t\to\lambda^{-3}t$. This symmetry can be used to derive a simple scaling formula that yields the Hamiltonians $H^{(n)}$ in the mKdV hierarchy, by the general scaling method [@Anc03] shown in the appendix. The formula is given by $$\label{mkdv-H-Q-expr}
H^{(n)}=\int_{-\infty}^{\infty}h^{(n)}\,dx, 
\quad
h^{(n)} = \tfrac{1}{2n+1} D\inv_x(vD_xQ^{(n)})$$ where the Hamiltonian density $h^{(n)}$ can be freely changed by the addition of a total $x$-derivative. Furthermore, the gradient recursion formula $Q^{(n)}= {\R^*}^n(v)$ can be used to convert the Hamiltonian expression into a recursion formula for $h^{(n)}$ itself, as follows. First, note expression can 