Hamiltonian structure comes directly from splitting the operator $\E=\E_1+\E_2$ in the bi-Hamiltonian equation for $P^{(0)} = -v_x$. This yields $c^{-1}\E(0)= c^{-1}\E_2(0) = -v_x$, since $\E_1(0) =0$. Hence, $$-v_x = c^{-1}\E(0)$$ holds with a trivial Hamiltonian.

Then, Magri’s theorem [@Mag] implies that the bi-Hamiltonian flow $$P^{(0)} = -v_x = \wtil\H(\delta \wtil H/\delta v) = c^{-1}\wtil\E(0)$$ defined by $x$-translations is a root flow for an integrable hierarchy of higher flows that are generated by the hereditary recursion operator $$\label{mkdv-peakon-Rop}
\wtil\R=\wtil\E\wtil\H\inv= D_x v D\inv_x v\Delta\inv .$$ Each flow in the resulting hierarchy $\wtil P^{(n)} = \wtil\R^n({-v_x})$, $n=1,2,\ldots$, yields an equation $v_t = \wtil P^{(n)}$ that has a bi-Hamiltonian structure $\wtil P^{(n)} = \wtil\H(\delta \wtil H^{(n)}/\delta v)=\wtil\E(\delta \wtil H^{(n-1)}/\delta v)$ with $\wtil H^{(0)}=\wtil H$. The gradients of the Hamiltonians $\wtil H^{(n)}$, $n=0,1,2,\ldots$, are generated by $\delta \wtil H^{(n)}/\delta v = \wtil\R^*{}^n(u) =\wtil Q^{(n)}$, $n=1,2,\ldots$, in terms of the adjoint recursion operator $$\label{mkdv-peakon-adRop}
\wtil\R^*=\wtil\H\inv\wtil\E=\Delta\inv v D\inv_x vD_x$$ where $\delta \wtil H^{(0)}/\delta v=\delta \wtil H/\delta v = u$. These Hamiltonian gradients are equivalently given by $$\label{mkdv-peakon-Q}
\delta \wtil H^{(n)}/\delta u = \wtil\K^n(v) = \Delta \wtil Q^{(n)}$$ in terms of the operator $$\label{mkdv-peakon-Kop}
\wtil\K=\Delta\wtil\R^*\Delta\inv=v D\inv_x vD_x\Delta\inv .$$ Moreover, as $\wtil\K$ is scaling homogeneous in terms of $u$, 