used to convert the Hamiltonian expression into a recursion formula for the Hamiltonian densities, similarly to the mKdV case. Specifically, substitution of $\wtil Q^{(n)}= \K \wtil Q^{(n-1)} =2n u^{-1} \tilde h^{(n-1)}$ into expression gives $$\label{mch-H-expr}
\tilde h^{(n)} =\tfrac{n}{n+1} uvD\inv_x\big(vD_x\Delta\inv(u^{-1}\tilde h^{(n-1)})\big)$$ modulo a total $x$-derivative. This provides an explicit recursion formula for all of the Hamiltonians in the mCH hierarchy, starting from the Hamiltonian density $\tilde h^{(0)}= \tfrac{1}{2}vu$.

All of the higher symmetries and higher Hamiltonian densities are found to be nonlocal expressions in terms of $u$, $u_x$, $v$, and $x$-derivatives of $v$. In particular, the first higher symmetry $\X^{(2)}=\wtil P^{(2)}\partial/\partial_v$ admitted by the mCH equation is given by $$\wtil P^{(2)} = \wtil\R\wtil P^{(1)} 
= \tfrac{1}{2} \big(v(uw-u_xw_x - \tfrac{1}{4}(u^2-u_x^2)^2)\big)_x$$ where $w$ is another potential which is defined by $$\Delta w = v(u^2-u_x^2) .$$ Likewise, the first higher Hamiltonian density $\tilde h^{(2)}$ admitted by the mCH equation is given by $$\tilde h^{(2)} =\tfrac{2}{3} uvD\inv_x\big(vD_x\Delta\inv(u^{-1}\tilde h^{(1)})\big)
= \tfrac{1}{12}vu(uw -u_xw_x-\tfrac{1}{4}(u^2-u_x^2)^2)$$ which yields the Hamiltonian $$\begin{aligned}
\tilde H^{(2)} & = \int_{-\infty}^{\infty} \tfrac{1}{12}vu(uw -u_xw_x-\tfrac{1}{4}(u^2-u_x^2)^2)\, dx
\\
& = \int_{-\infty}^{\infty} \tfrac{1}{12}( u^3w +\tfrac{3}{2} uu_x^2w - \tfrac{1}{2} u_x^3 w_x +\tfrac{1}{4}u^2u_x^2(u^2+u_x^2) -\tfrac{1}{4} u^6 -\tfrac{1}{10}u_x^6  )\, dx
\end{aligned}$$ modulo boundary terms.

Recursion operator method
-------------------------

From the splitting of the 