the Hamiltonian structure $$\label{nls-evenroot-flow}
P^{(0)} = \s\inum v 
=\I(\delta E^{(0)}/\delta\bar v)$$ while the root flow in the NLS hierarchy of odd flows has the bi-Hamiltonian structure $$\label{nls-oddroot-flow}
P^{(1)} = -v_x 
=-\I(\delta H^{(0)}/\delta\bar v) 
= \H(\delta E^{(0)}/\delta\bar v) .$$ Both of these root flows possess another Hamiltonian structure if, similarly to the mKdV case, $D\inv_x(0)$ is redefined by the addition of some non-zero constant, so that $D\inv_x(0)=c\neq 0$. Then the relations $\H(0)=-\inum v D\inv_x(0)$ and $\D(0)=\s(\inum v -v_x)D\inv_x(0)$ yield the Hamiltonian structures $$\label{nls-rootflows-othstruct}
-P^{(0)} = \s c^{-1}\H(0),
\quad
P^{(1)} = c^{-1}(\H(0) \s \D(0))$$ where the corresponding Hamiltonians are trivial.

Each higher flow in the combined NLS hierarchy corresponds to a higher symmetry operator $\X^{(n)}=P^{(n)}\partial_v$, $n=1,2,\ldots$, all of which represent infinitesimal symmetries that are admitted by every equation $v_t= P^{(n)}$, $n=1,2,\ldots$, in the combined hierarchy. The Hamiltonians $E^{(n)}$, $H^{(n)}$, $n=0,1,2,\ldots$, of the even and odd flows in the combined hierarchy correspond to conservation laws $\frac{d}{dt} H^{(n)}=\frac{d}{dt} E^{(n)}=0$, all of which hold for every equation $v_t= P^{(n)}$, $n=1,2,\ldots$, in the combined hierarchy. All of the higher symmetries and higher Hamiltonian densities are local expressions in terms of $v$ and its $x$-derivatives.

Similarly to the mKdV case, the general scaling method [@Anc03] shown in 