-\I(\delta H^{(n)}/\delta\bar v)
= \H(\delta E^{(n)}/\delta\bar v)
=-\D(\delta H^{(n-1)}/\delta\bar v)
= \E(\delta E^{(n-1)}/\delta\bar v), 
\quad
n=1,2,\ldots$$ where $$\label{nls-Eop}
\begin{aligned}
\E=-\R^2\H & 
=-\big( D_x^3 +D_xvD\inv_x\Re\bar vD_x 
+ \inum D_x^2vD\inv_x\Im\bar v + \inum vD\inv_x\Im\bar v D_x^2
\\&\qquad\qquad
+\tfrac{1}{2}\inum v(|v|^2 D\inv_x\Im\bar v + D\inv_x |v|^2 \Im\bar v) \big)
\end{aligned}$$ is a fourth Hamiltonian operator which is mutually compatible with $\I$, $\H$, $\D$. In particular, the first higher flow in this hierarchy is an integrable $U(1)$-invariant version of the mKdV equation given by $$\label{hirota-eqn}
v_t = \R(\s\inum(\tfrac{1}{2}|v|^2 v +v_{xx})) = -\R^2(-v_x) 
= -v_{xxx}-\tfrac{3}{2}|v|^2v_x$$ which is the Hirota equation [@Hir1973]. It possesses the quad-Hamiltonian structure $$\label{hirota-quadHamil}
v_t = -v_{xxx}-\tfrac{3}{2}|v|^2v_x
= -\I(\delta H^{(1)}/\delta\bar v) 
= \H(\delta E^{(1)}/\delta\bar v) 
= -\D(\delta H^{(0)}/\delta\bar v) 
= \E(\delta E^{(0)}/\delta\bar v)  .$$ Its recursion operator is given by the NLS squared recursion operator .

In the NLS hierarchy, the real reductions of $\H$ and $\E$ match the mKdV Hamiltonian operators –, while the Hamiltonians $E^{(n)}$, $n=0,1,2,\ldots$ match the mKdV Hamiltonians , , multiplied by a factor of $2$, with the real reduction of the variational derivative $\delta/\delta\bar v = \tfrac{1}{2}(\delta/\delta\Re v +\inum \delta/\delta\Im v)$ having a compensating factor of $1/2$.

The root flow in the NLS hierarchy of even flows has 