reality condition $\bar v=v$ is given by the negative of the mKdV recursion operator . Consequently, for comparison with the mKdV hierarchy, the most natural way to generate higher flows using the NLS recursion operator is by dividing the NLS hierarchy into even and odd flows $$\begin{aligned}
P^{(2n)} &=(-\R^2)^n(\s\inum v),
\quad 
n=1,2,\ldots
\label{nls-even-flows}
\\
P^{(2n+1)} &=(-\R^2)^n({-v_x}),
\quad
n=1,2,\ldots
\label{nls-odd-flows}\end{aligned}$$ which are related by $$P^{(2n+1)} = \R P^{(2n)} .$$ Then all of the odd flows will admit a real reduction that yields a corresponding flow in the mKdV hierarchy, whereas all of the even flows do not possess a real reduction.

Every flow in the NLS hierarchy – inherits a tri-Hamiltonian structure, due to Magri’s theorem. For the even flows, this structure is given by $$\label{nls-even-triHam}
P^{(2n)} = \I(\delta E^{(n)}/\delta\bar v)
= \H(\delta H^{(n-1)}/\delta\bar v)
=\D(\delta E^{(n-1)}/\delta\bar v),
\quad
n=1,2,\ldots$$ where the gradients of the Hamiltonians are generated by $$\begin{aligned}
\delta E^{(n)}/\delta\bar v & = (-\R^*{}^2)^n(v) = Q^{(2n)}, 
\quad
n=0,1,2,\ldots
\\
\delta H^{(n)}/\delta\bar v & = -(-\R^*{}^2)^n(\s\inum v_x) = Q^{(2n+1)}= -\R^*Q^{(2n)},
\quad
n=0,1,2,\ldots\end{aligned}$$ and where $\R^*$ is the adjoint recursion operator $$\label{nls-adRop}
\R^*=\I\inv\H=\s(\inum D_x -vD\inv_x \Im\bar v) .$$ The odd flows have a similar but enlarged Hamiltonian structure $$\label{nls-odd-triHam}
P^{(2n+1)} = 