---
abstract: 'We study nonlocal reductions of coupled equations in $1+1$ dimensions of the Heisenberg ferromagnet type. The equations under consideration are completely integrable and have a Lax pair related to a linear bundle in pole gauge. We describe the integrable hierarchy of nonlinear equations related to our system in terms of generating operators. We present some special solutions associated with four distinct discrete eigenvalues of scattering operator. Using the Lax pair diagonalization method, we derive recurrence formulas for the conserved densities and find the first two simplest conserved densities.'
author:
- |
    R. Myrzakulov$^1$, G. Nugmanova$^1$, T. Valchev$^2$, K. Yesmakhanova$^1$\
    $^1$ Department of General & Theoretical Physics,\
    Eurasian National University, Astana, 010008, Kazakhstan\
    $^2$ Institute of Mathematics and Informatics,\
    Bulgarian Academy of Sciences, Acad. G. Bonchev Str.,\
    1113 Sofia, Bulgaria\
    E-mails: tiv@math.bas.bg, rmyrzakulov@gmail.com,\
    nugmanovagn@gmail.com, kryesmakhanova@gmail.com
title: Nonlocal Reductions of a Generalized Heisenberg Ferromagnet Equation
---

Introduction
============

In [@yantih; @yantih2], the coupled system of nonlinear evolution equations (NLEEs) $$\begin{split}
&\rmi u_t + u_{xx}+(\varepsilon uu^*_x +  vv^*_x)u_x
+ (\varepsilon uu^*_x + vv^*_x)_x u  =  0\, , \qquad \varepsilon^2 = 1 \, ,\\
&\rmi v_t + v_{xx}+(\varepsilon uu^*_x + vv^*_x)v_x
+ (\varepsilon uu^*_x + vv^*_x)_x v  = 0\, ,\qquad \rmi=\sqrt{-1}\,
\end{split} \label{ghf}$$ and the corresponding auxiliary spectral problem were introduced and studied. Above, subscripts stand for partial differentiation; $*$ denotes complex conjugation; $|z| = \sqrt{zz^*}$, $z\in\bbc$ and the complex valued functions $u$ and $v$ obey the constraint: $$\varepsilon |u|^2 + |v|^2 = 1 .
\label{constr_0}$$ System (\[ghf\]) with $\varepsilon = 1$ (so-called Hermitian reduction) was extensively studied in [@gmv; @jgsp; @side9; @yan; @yanvil]. It represents an integrable generalization of the classical Heisenberg ferromagnet equation $$\mathbf{S}_t = \mathbf{S}\times \mathbf{S}_{xx},\qquad \mathbf{S}^2 = 1$$ for the spin vector $\mathbf{S} = (S^{\,1},S^{\,2},S^{\,3})$, see [@BoPo90; @book; @blue-bible]. Its zero curvature representation is determined by the following Lax pair: $$\begin{aligned}
L(\lambda) & = & \rmi\partial_x - \lambda S, \qquad \lambda\in\bbc,\qquad
S = \left(\begin{array}{ccc}
0 & u & v\\ 
\varepsilon u^* & 0 & 0 \\
v^* & 0 & 0\end{array}\right), \label{ghf_lax1}\\
A(\lambda) &=& \rmi\partial_t + \lambda A_1 + \lambda^2 A_2,
\qquad A_2  = \left(\begin{array}{ccc}
- 1/3 & 0 & 0 \\ 0 & 2/3 - \varepsilon|u|^2 & -\varepsilon u^*v \\
0 & - v^*u & 2/3 -|v|^2
\end{array}\right),\label{ghf_lax2}\\
A_1 & = & \left(\begin{array}{ccc}
0 & a & b\\ 
\varepsilon a^* & 0 & 0 \\
b^* & 0 & 0\end{array}\right),\quad
\begin{array}{ccc} 
a & = & - \rmi u_x - \rmi\left(\varepsilon uu^*_x + vv^*_x\right)u\\
b & = & - \rmi v_x - \rmi\left(\varepsilon uu^*_x + vv^*_x\right)v
\end{array}\label{ghf_lax3}.\end{aligned}$$

Our main purpose here is to study a nonlocal counterpart of (\[ghf\]), namely the following system of NLEEs $$\begin{split}
\rmi u_t(x,t) + u_{xx}(x,t) + [B(x,t)u(x,t)]_x = 0,\\\
\rmi v_t(x,t) + v_{xx}(x,t) + [B(x,t)v(x,t)]_x = 0,
\end{split}
\label{ghf_a}$$ where $B(x,t) = \varepsilon u(x,t)u^*_x(-x,t) + v(x,t)v^*_x(-x,t)$. This system is completely integrable and it has a Lax representation. In the sections to follow we shall describe the integrable hierarchy of NLEE related to (\[ghf\_a\]), some particular solutions and discuss the corresponding integrals of motion.

The paper itself is organized as follows. Next section introduces the main object of study and its Lax pair. In Section \[sec\_hier\] we describe the hierarchy of NLEEs related to (\[ghf\_a\]) in terms of recursion operators. Another issue of fundamental importance concerns the solutions of (\[ghf\_a\]). We shall show here how one can construct particular solutions to (\[ghf\_a\]) in the simplest case of constant background. In doing this, we shall make use of dressing method that seems to be suitable for the system of equations we are interested in.

Section \[consdens\] is dedicated to conserved densities. In order to derive the conserved densities of (\[ghf\_a\]) we shall apply the method of Lax pair diagonalization [@DrSok*85].

Last section contains some concluding remarks.

Lax Representation {#laxpair}
==================

Let us consider the following Lax pair:

$$\begin{aligned}
L(\lambda) & = & \rmi\partial_x - \lambda S, \quad \lambda\in\bbc,\quad
S = \left(\begin{array}{ccc}
0 & u(x,t) & v(x,t)\\ 
\varepsilon u^*(-x,t) & 0 & 0 \\
v^*(-x,t) & 0 & 0\end{array}\right), \quad \varepsilon = \pm 1 ,\label{lax1}\\
A(\lambda) &=& \rmi\partial_t + \lambda A_1 + \lambda^2 A_2,\quad
A_1 = \left(\begin{array}{ccc}
0 & a(x,t) & b(x,t)\\ 
-\varepsilon a^*(-x,t) & 0 & 0 \\
-b^*(-x,t) & 0 & 0\end{array}\right),\nonumber\\
a(x,t) & = & - \rmi u_x(x,t) - \rmi B(x,t)u(x,t),\quad B(x,t) = \varepsilon u(x,t)u^*_x(-x,t) + v(x,t)v^*_x(-x,t)
\nonumber\\
b(x,t) & = & - \rmi v_x(x,t) - \rmi B(x,t)v(x,t)\label{lax2}\\
A_2  & = & \frac{2}{3}\openone - S^2 = \left(\begin{array}{ccc}
-1/3 & 0 & 0 \\ 0 & 2/3 - \varepsilon u^*(-x,t)u(x,t) & - \varepsilon u^*(-x,t)v(x,t)\\
0 & - v^*(-x,t)u(x,t) & 2/3 - v^*(-x,t)v(x,t)
\end{array}\right)\label{lax3}.\end{aligned}$$

The function $u(x,t)$ and $v(x,t)$ are not arbitrary but are subject to the constraint: $$\varepsilon u(x,t)u^*(-x,t) + v(x,t)v^*(-x,t) = 1.
\label{constr}$$ We note that (\[constr\]) implies that the matrix $S$ satisfies: $$S^3 = S.
\label{s_rel}$$

The compatibility condition of those Lax operators leads to the following nonlinear evolution equations: $$\begin{split}
\rmi u_t(x,t) + u_{xx}(x,t) + [B(x,t)u(x,t)]_x = 0,\\\
\rmi v_t(x,t) + v_{xx}(x,t) + [B(x,t)v(x,t)]_x = 0.
\end{split}
\label{ghf_nonloc}$$ The form of the matrix coefficients in (\[lax1\]), (\[lax2\]) and (\[lax3\]) implies that the Lax operators are subject to the following symmetry conditions: $$\begin{aligned}
HS(x,t)H &=& - S(x,t) ,\quad HA_2(x,t)H = A_2(x,t),\quad HA_{1}(x,t)H = -A_1(x,t),\label{red1}\\
QS^{\dag}(-x,t)Q  &=& S(x,t) ,\quad QA_2^{\dag}(-x,t)Q = A_2(x,t), \quad 
QA_1^{\dag}(-x,t)Q = - A_1(x,t)\label{red2}\end{aligned}$$ where $H=\diag(-1,1,1)$, $Q = \diag(1,\varepsilon,1)$. These symmetries are due to the action of group $\bbz_2\times \bbz_2$ on the space of the fundamental solutions $\{\Psi(x,t,\lambda)\}$ to the spectral problem $$\rmi\partial_x\Psi(x,t,\lambda) - \lambda S(x,t)\Psi(x,t,\lambda) = 0.$$ Indeed, one can easily check that the following action $$\begin{aligned}
\Psi(x,t,\lambda)&\to& \tilde{\Psi}(x,t,\lambda) = H\Psi(x,t,-\lambda)H,\label{psired1}\\
\Psi(x,t,\lambda) &\to& \tilde{\Psi}(x,t,\lambda) = Q\hat{\Psi}^{\dag}(-x,t,-\lambda^*)Q\label{psired2}\end{aligned}$$ leads to relations (\[red1\]) and (\[red2\]). Above the symbol $\hat{}$ stands for matrix inverse, i.e we have $\hat{\Psi}(x,t,\lambda) = [\Psi(x,t,\lambda)]^{-1}$. In this case it is said that the Lax pair and the corresponding NLEE are $\bbz_2\times \bbz_2$-reduced [@mikh1; @mikh2].

Integrable Hierarchies {#sec_hier}
======================

In this section, we shall describe the hierarchy of matrix integrable NLEEs associated with equations (\[ghf\_nonloc\]). In doing this, we shall follow ideas and methods discussed in [@book; @side9].

Let us consider the general flow Lax pair: $$\begin{split}
L(\lambda) :=& \rmi\partial_x - \lambda S,\\
A(\lambda) :=& \rmi\partial_t + \sum^N_{j=1}\lambda^j A_j,\qquad N\geq 2
\label{ghf_lax_gen}
\end{split}$$ which is subject to reduction conditions (\[red1\]) and (\[red2\]). The matrix $S(x,t)$ has the same form as in (\[lax1\]) and it is assumed to obey (\[s\_rel\]), i.e. constraint (\[constr\]) holds true again. Then the zero curvature condition $[L(\lambda), A(\lambda)] = 0$ of the Lax pair (\[ghf\_lax\_gen\]) gives rise to the following set of recurrence relations: $$\begin{aligned}
&&  [S, A_N] = 0 , \label{rec_np1}\\
&& \ldots \nonumber\\
&& \rmi\partial_xA_k - [S,A_{k-1}] = 0,
\qquad k=2,\ldots, N ,\label{rec_k}\\
&& \ldots \nonumber\\
&& \partial_xA_1 + \partial_t S = 0.\label{rec_1}\end{aligned}$$ Each solution to these equations leads to a member of the integrable hierarchy of NLEEs. The analysis of (\[rec\_np1\])–(\[rec\_1\]) resembles very much the one we have in the case of (\[ghf\]). Following [@side9; @yantih],we introduce the splitting $$A_j = A^{\rm a}_j + A^{\rm d}_j\ , \qquad j=1,\ldots,N
\label{a_split}$$ of the matrix coefficients of $A$ into a $S$-commuting term $A^{\rm d}_j$ and some remainder $A^{\rm a}_j$. From (\[s\_rel\]) it follows that $S(x,t)$ is a diagonalizable matrix with eigenvalues: $\pm 1, 0$. Hence $\ad_S$ is diagonalizable as well with eigenvalues $0,\pm 1, \pm 2$. The above splitting simply says that $ A^{\rm d}_j$ and $ A^{\rm a}_j$ belong to the zero eigenspace of $\ad_S$ and the direct sum of all the nonzero eigenspaces respectively. Consequently, the above splitting is unique and the operator $\ad_S^{-1}$ is properly defined on $ A^{\rm a}_j$, namely we have $$\ad^{-1}_{S} = \frac{1}{4}\left(5\ad_{S} - \ad^3_{S}\right).$$

Symmetry conditions (\[red1\]) and (\[red2\]) require that we have $$A_j\in \mathfrak{sl}^{\sigma}(3,\bbc) := \{X\in \mathfrak{sl}(3,\bbc); HXH = (-1)^{\sigma}X\},
\quad j \equiv \sigma \quad(\mod 2), \quad \sigma = 0, 1$$ for the coefficients of the second Lax operator, see the comments in [@yantih]. For our purposes it will be enough to take $$A^{\rm d}_{j} = \begin{cases}
a_{j} S_1,& j \equiv 0 \quad(\mod 2)\\ a_{j} S,& j\equiv 1 \quad (\mod 2) 
\end{cases}
\label{a_j_diag}$$ where $$S_1 := S^{\,2} - \frac{2}{3}\openone_{3}$$ and $a_{j}$, $j=1,\ldots,N$ are some scalar functions to be determined in such a way that the recurrence relations are satisfied.

Equation (\[rec\_np1\]) means that $A^{\rm a}_N = 0$ and in accordance with our previous assumption, for the highest degree coefficient we may take: $$A_{N} = \begin{cases} 
c_NS ,& N \equiv 1 \quad (\mod 2) \\ c_N S_1 ,& N \equiv 0 \quad (\mod 2)
\end{cases},\qquad c_N\in\bbr.$$

Let us consider now recurrence relation (\[rec\_k\]). After substituting (\[a\_split\]) into (\[rec\_k\]) and taking into account that $$(S_x)^{\rm d} =  (S_{1,x})^{\rm d} = 0,$$ we derive the following two relations $$\left(\partial_xA^{\rm a}_k\right)^{\rm d} = - \begin{cases} 
\partial_xa_k S ,& k \equiv 1 \quad (\mod 2) \\ \partial_xa_k S_1 ,& k \equiv 0 \quad (\mod 2) \end{cases}\; ,\qquad k=2,\ldots,N
\label{rec_k_d}$$ for the $S$-commuting terms and $$\left[S, A^{\rm a}_k\right] - \rmi\left(\partial_xA^{\rm a}_k\right)^{\rm a} =
\begin{cases} 
\rmi a_k S_x ,& k \equiv 1 \quad (\mod 2) \\ \rmi a_k S_{1,x} ,& k \equiv 0 \quad (\mod 2) \end{cases}
\label{rec_k_a}$$ for the terms not commuting with $S$. In order to solve (\[rec\_k\_d\]), we also make use of the normalization conditions: $$\tr S^2 = 2\, ,\qquad \tr S^2_1 = \frac{2}{3}\, .$$ As a result, we obtain the following expression for $a_k$ $$a_k = c_k - \begin{cases} 
\frac{1}{2}\,\partial^{-1}_x\tr\left[S(\partial_xA_k)^{\rm d}\right] ,& k \equiv 1 \quad (\mod 2) \\ \frac{3}{2}\,\partial^{-1}_x\tr\left[S_1(\partial_xA_k)^{\rm d}\right] ,& k \equiv 0 \quad (\mod 2) \end{cases}    
\label{a_k}$$ where the symbol $\partial^{-1}_x$ stands for any right inverse of the operator of partial differentiation in variable $x$ and $c_k\in\bbr$ is an integration constant. After substituting (\[a\_k\]) into (\[rec\_k\_a\]), we obtain $$A^{\rm a}_{k-1} = \begin{cases}
\Lambda A^{\rm a}_{k} + \rmi c_k\ad^{-1}_{S}S_{1,x}\, ,&
k\equiv 0 \quad (\mod 2)\\
\Lambda A^{\rm a}_{k} + \rmi c_k\ad^{-1}_{S}S_{x}\, ,&
k\equiv 1 \quad (\mod 2) 
\end{cases}
\label{A_km1}$$ where $$\Lambda := \rmi\ad^{-1}_{S}\left\{ [\partial_x(.)]^{\rm a} - \frac{S_{x}}{2}
\partial^{-1}_x\tr\left[S(\partial_x(.))^{\rm d}\right] - \frac{3 S_{1,x}}{2}
\partial^{-1}_x\tr\left[S_1(\partial_x(.))^{\rm d}\right]\right\}.
\label{Lambda}$$ The $\Lambda$ operator as introduced above acts on the $S$-non commuting part of $A_j$ only. However, one can extend the action of $\Lambda$ on the $S$-commuting part as well by defining its action to be $$\Lambda S :=  \rmi \ad^{-1}_{S} S_{x}\, ,\qquad
\Lambda S_{1} :=  \rmi \ad^{-1}_{S} S_{1,x}\,.$$ Taking into account (\[A\_km1\]), (\[Lambda\]) for an arbitrary member of the integrable hierarchy under consideration, we have $$\rmi\ad^{-1}_{S} S_{t}  + \sum_{k}c_{2k}\Lambda^{2k}S_1
+ \sum_{k}c_{2k-1}\Lambda^{2k-1} S = 0.
\label{int_hier}$$ The operator $\Lambda^2$ is called recursion operator of the above hierarchy of NLEEs. It can be checked that (\[int\_hier\]) gives (\[ghf\_nonloc\]) after setting $N=2$, $c_2 = -1$ and $c_1 = 0$. Thus, it is the simplest nontrivial member of the family (\[int\_hier\]).

Special Solutions {#solutions}
=================

In this section, we shall construct special solutions to the system (\[ghf\_nonloc\]) obeying the following boundary condition: $$\lim_{x\to\pm\infty}u(x,t) = 0 \, ,\qquad \lim_{x\to\pm\infty}v(x,t) = 1
\label{triv_back}$$ that is compatible with (\[constr\]). Our approach to obtain particular solutions will be based on Zakharov-Shabat’s dressing method [@ZS; @zakh-mikh] as implemented in [@yantih2] for the case linear bundles in pole gauge.

Let us consider the Lax pair $$\begin{aligned}
L_0(\lambda) & = & \rmi\partial_x - \lambda S^{(0)}, \qquad 
S^{(0)}(x,t) = \left(\begin{array}{ccc}
0 & u_0(x,t) & v_0(x,t)\\ 
\varepsilon u^*_0(-x,t) & 0 & 0 \\
v^*_0(-x,t) & 0 & 0\end{array}\right), \label{lax1_bare}\\
A_0(\lambda) & = & \rmi\partial_t + \lambda A^{(0)}_1 + \lambda^2 A^{(0)}_2 ,
\qquad \lambda\in\bbc\label{lax2_bare}\end{aligned}$$ where $(u_0,v_0)$ are two known functions and the coefficients of the second Lax operator are expressed in terms of $(u_0,v_0)$ in the same way as given in (\[lax2\]) and (\[lax3\]). In other words, $(u_0,v_0)$ are assumed to satisfy (\[ghf\_nonloc\]) and are subject to conditions (\[constr\]) and (\[triv\_back\]). Let $\Psi_0$ be an arbitrary fundamental solution to the auxiliary linear problem: $$L_0(\lambda) \Psi_0(x,t,\lambda) =  0 \, .
\label{bare1}$$ Due to the compatibility of $L_0$ and $A_0$, $\Psi_0$ also fulfills the linear problem: $$A_0(\lambda)\Psi_0(x,t,\lambda) = \Psi_0(x,t,\lambda) f(\lambda)
\label{bare2}$$ where $$f(\lambda) := \lim_{x\to\pm\infty}\hat{g}(x,t)\left[\lambda A^{(0)}_1(x,t) + \lambda^2A^{(0)}_2(x,t)\right]g(x,t)
= - \lambda^2 \diag(1,-2,1)/3
\label{disp_law}$$ is the dispersion law of NLEE. Above, $$g(x,t) = \frac{\sqrt{2}}{2}\left(\begin{array}{ccc}
1 & 0 & -1 \\ \varepsilon u^*(-x,t) & \sqrt{2} v(x,t) & \varepsilon u^*(-x,t)\\
v^*(-x,t) & -\sqrt{2} u(x,t)  & v^*(-x,t)
\end{array}\right), \qquad \hat{g}(x,t) = [g(x,t)]^{-1}$$ is the gauge transform diagonalizing $S^{(0)}$, see [@yantih; @yantih2] for more explanations. The linear problems (\[bare1\]) and (\[bare2\]) are called bare linear problems and their fundamental solutions will be called bare fundamental solutions respectively. We shall denote the set of all bare fundamental solutions by $\mathcal{F}_0$.

Now, let us apply the dressing transform $$\mathcal{G}:\mathcal{F}_0 \to \mathcal{F}_1 = \mathcal{G}\mathcal{F}_0
:= \{\mathcal{G}\Psi_0| \Psi_0\in\mathcal{F}_0\}
\label{drestrans}$$ where $\mathcal{G}(x,t,\lambda)$ is a $3\times 3$-matrix, such that $\det \mathcal{G} = 1$. Since the Lax operators are transformed through: $$L_0 \to L_1 := \mathcal{G}L_0\hat{\mathcal{G}}\, ,\qquad A_0 \to A_1 := \mathcal{G}A_0\hat{\mathcal{G}}\, ,
\label{lax_dres}$$ the operators $L_1$ and $A_1$ commute as well. We shall require that the auxiliary linear problems remain covariant under (\[drestrans\]), i.e. we have $$L_1(\lambda) \Psi_1(x,t,\lambda) = 0\, , \qquad A_1(\lambda)\Psi_1(x,t,\lambda) 
= \Psi_1(x,t,\lambda)f(\lambda)\,,\qquad \Psi_1\in\mathcal{F}_1
\label{dres}$$ where $L_1$ and $A_1$ read $$\begin{aligned}
L_1(\lambda) & = & \rmi\partial_x - \lambda S^{(1)}, \qquad 
S^{(1)} = \left(\begin{array}{ccc}
0 & u_1(x,t) & v_1(x,t)\\ 
\varepsilon u^*_1(-x,t) & 0 & 0 \\
v^*_1(-x,t) & 0 & 0\end{array}\right), \label{lax1_dres}\\
A_1(\lambda) & = & \rmi\partial_t + \lambda A^{(1)}_1 + \lambda^2 A^{(1)}_2. \label{lax2_dres}\end{aligned}$$ Above, $u_1$ and $v_1$ are some new (unknown) functions satisfying the same NLEE. After comparing (\[bare1\]) and (\[bare2\]) with (\[dres\]), we see that $\mathcal{G}$ solves the following system of linear partial differential equations: $$\begin{aligned}
\rmi\partial_x \mathcal{G} & - & \lambda \left(S^{(1)}\mathcal{G} - \mathcal{G}S^{(0)}\right) = 0\, ,
\label{g_pde1}\\
\rmi\partial_t \mathcal{G} & + & \sum_{k=1,2}\lambda^k \left( A^{(1)}_k \mathcal{G}
- \mathcal{G} A^{(0)}_k\right) = 0\,.
\label{g_pde2}\end{aligned}$$ After dividing (\[g\_pde1\]) by $\lambda$ and setting $|\lambda|\to\infty$, we derive the following interrelation: $$S^{(1)} = \mathcal{G}_{\infty}S^{(0)}\hat{\mathcal{G}}_{\infty},\qquad
\mathcal{G}_{\infty}(x,t) := \lim_{|\lambda|\to\infty}\mathcal{G}(x,t,\lambda)\, .
\label{s10_eq}$$ provided $\mathcal{G}$ as well as its derivatives in $x$ and $t$ are regular when $|\lambda|\to \infty$. Since $S^{(0)}$ is determined by $(u_0,v_0)$ and $S^{(1)}$ is determined by $(u_1,v_1)$, the above relation allows us to construct another solution of our system starting from a known one.

The $\bbz_2\times\bbz_2$ action on $\mathcal{F}_0$ and $\mathcal{F}_1$ implies the dressing factor obeys the symmetry relations [@yantih2]: $$\begin{aligned}
H\mathcal{G}(x,t,-\lambda)H &=& \mathcal{G}(x,t,\lambda)\,,\label{g_red1}\\
Q\mathcal{G}^{\dag}(-x,t,-\lambda^*)Q &=& \hat{\mathcal{G}}(x,t,\lambda)\, .
\label{g_red2}\end{aligned}$$ A simple ansatz for dressing factor is given by the meromorphic function: $$\mathcal{G}(x,t,\lambda) = \openone + \lambda\sum_{j}\left[ \frac{ B_j(x,t)}{\mu_j(\lambda - \mu_j)}
+ \frac{HB_j(x,t)H}{\mu_j(\lambda + \mu_j)}\right],
\qquad \mu_j\in\bbc\backslash\{0\}\, .
\label{dress_fac}$$

In order to find $L_1 $ and $A_1$ through (\[s10\_eq\]), we need to know $B_j(x,t)$. The algorithm to find the residues of (\[dress\_fac\]) consists in two steps. In the first step, one considers the identity $\mathcal{G}\hat{\mathcal{G}}=\openone$ which gives rise to a set of algebraic relations for $B_j$. Those always imply that the residues of $\mathcal{G}$ are some degenerate matrices which could be decomposed as follows: $$B_j(x,t) = X_j(x,t)F^T_j(x,t)
\label{res_decomp}$$ where $X_j(x,t)$ and $F_j(x,t)$ are rectangular matrices of certain rank and the superscript $T$ stands for matrix transposition. After substituting (\[res\_decomp\]) into the algebraic relations, we are able to express $X_j$ through $F_j$. The factors $F_j$ are determined from (\[g\_pde1\]) and (\[g\_pde2\]).

In that second step equation (\[g\_pde1\]) can be shown to lead to the following result $$F^T_j(x,t) = F^T_{j,0}(t)\hat{\Psi}_0(x,t,\mu_j).$$ This allows us to construct $F_j$ through some bare fundamental solution defined in the vicinity of $\mu_j$ and arbitrary $x$-independent matrices $F_{j,0}$. The dependence of $F_{j,0}$ on $t$ is determined by (\[g\_pde2\]). Thus, to recover the time dependence in all formulas we may use the rule below: $$F^T_{j,0}(t)\quad\to\quad F^T_{j,0}(t)\rme^{-\rmi f(\mu_j)t}$$ where $f(\lambda)$ is the dispersion law of the NLEE, see (\[disp\_law\]). We shall demonstrate that procedure in the case when the poles of dressing factor are all generic complex numbers.

Let us consider the case when the poles $\mu_j$ of (\[dress\_fac\]) are complex numbers obeying $\mu^2_j\notin\bbr$ for all $j$. From the symmetry condition (\[g\_red2\]), we immediately deduce that $$\hat{\mathcal{G}}(x,t,\lambda) = \openone + \lambda\sum_i\left[\frac{QB^{\dag}_i(-x,t)Q}
{\mu^*_i(\lambda + \mu^*_i)} +  \frac{QHB^{\dag}_i(-x,t)HQ}
{\mu^*_i(\lambda - \mu^*_i)}\right].
\label{g_inv1}$$ Thus, the dressing factor and its inverse have poles located at different points.

Let us consider the identity $\mathcal{G}(x,t,\lambda)\hat{\mathcal{G}}(x,t,\lambda)=\openone$. After evaluating the residue of $\mathcal{G}\hat{\mathcal{G}}$ at $\mu^*_i$ we easily derive $$\left[\openone + \mu^*_i\sum_j\left(\frac{B_j(x,t)}{\mu_j(\mu^*_i - \mu_j)}
+ \frac{HB_j(x,t)H}{\mu_j(\mu^*_i + \mu_j)}\right)\right]QHB^{\dag}_i(-x,t)HQ = 0\, .   
\label{algsys1}$$ Evaluation of the residues at $\pm\mu_i$ and $-\mu_i^*$ leads to equations that can easily be reduced to (\[algsys1\]), thus giving us no new constraints.

It is seen from (\[algsys1\]) that each $B_i$ should be a degenerate matrix, hence it can be factored $$B_i(x,t)=X_i(x,t)F^T_i(x,t)
\label{b_fac}$$ where $X_i$ and $F_i$ are two rectangular matrices. After substituting (\[b\_fac\]) into (\[algsys1\]), we have $$QHF^*_i(-x) = \mu^*_i\sum_j\left(X_j(x)\frac{F^T_j(x)QHF^*_i(-x)}{\mu_j(\mu_j - \mu^*_i)}
- HX_j(x)\frac{F^T_j(x)QF^*_i(-x)}{\mu_j(\mu^*_i + \mu_j)}\right)
\label{matr_sys1}$$ for the factors $X_j$. Solving that linear system, allows one to express $X_j$ through $F_j$. This is especially easy when the $\mathcal{G}$ has a single pair of poles: $\mu$ and $-\mu$. Assuming $X$ and $F$ are column-vectors, we immediately get: $$X(x) =  \left(\frac{\mu^*F^T(x)QHF^*(-x)}{\mu(\mu - \mu^*)}
- \frac{\mu^*F^T(x)QF^*(-x)}{\mu(\mu + \mu^*)}H\right)^{-1}QHF^*(-x) . 
\label{XF_sys}$$

Let us return now to the general case. The factors $F_j$ can be found from differential equation (\[g\_pde1\]). For that purpose we rewrite (\[g\_pde1\]) in the following way $$\lambda S^{(1)}= \rmi\partial_x \mathcal{G}\hat{\mathcal{G}} + \lambda\mathcal{G}S^{(0)}\hat{\mathcal{G}}$$ and calculate the residues of its right-hand side at $\lambda=\mu_i$. After taking into account (\[matr\_sys1\]), we obtain the linear differential equation $$\rmi\partial_x F^T_i + \mu_i F^T_iS^{(0)} = 0 
\label{F_res}$$ which is easily solved to give $$F^T_i(x,t) = F^T_{i, 0}(t)\hat{\Psi}_0(x,t,\mu_i)\, .
\label{f_psi0}$$ Above, $F_{i,0}$ are “integration constant” matrices which depend on $t$ however. The $t$-dependence of $F_{i,0}$ can be determined from (\[g\_pde2\]) after it is rewritten as follows: $$\sum_{k=1,2}\lambda^k A^{(1)}_k = - \rmi\partial_t \mathcal{G}\hat{\mathcal{G}}
+ \mathcal{G} \sum_{k=1,2}\lambda^k A^{(0)}_k \hat{\mathcal{G}}\, .
\label{g_pde2_2}$$ Evaluation of the residues of both hand sides of (\[g\_pde2\_2\]) for $\lambda=\mu_i$ gives $$\rmi\partial_tF^T_i - F^T_i\sum_{k=1,2}\mu^k_i A^{(0)}_k = 0\, .
\label{fj_t}$$ After substituting (\[f\_psi0\]) into (\[fj\_t\]), we derive a linear differential equation for $F_{i,0}$, namely $$\rmi\partial_t F^T_{i,0} - F^T_{i,0} f(\mu_i) = 0
\label{fj0_t}$$ where $f(\lambda)$ is the dispersion law of the NLEE. Equation (\[fj0\_t\]) is easily solved and allows us to state the rule: in order to recover the $t$-dependence in all formulas, one has to make the following substitution $$F^T_{i,0}\quad\to\quad F^T_{i,0}\rme^{-\rmi f(\mu_i)t} .
\label{f_j0_evol}$$

Now, let us pick up $$S^{(0)} = \left(\begin{array}{ccc}
0 & 0 & 1 \\ 0 & 0 & 0 \\
1 & 0 & 0
\end{array}\right).
\label{seed}$$ as a bare solution. That obvious choice satisfies (\[constr\]) and (\[triv\_back\]). It can be proven that the bare scattering operator in this case has no discrete eigenvalues.

For a bare fundamental solution we shall choose $$\Psi_0(x,\lambda) = \left(\begin{array}{ccc}
\cos{\lambda x} & 0 & - \rmi\sin{\lambda x} \\ 0 & 1 & 0 \\
- \rmi\sin{\lambda x} & 0 & \cos{\lambda x}
\end{array}\right). 
\label{psi_0}$$ That fundamental solution is invariant with respect to both reductions (\[psired1\]) and (\[psired2\]) which makes it rather convenient for the calculations to follow.

From that point on, we shall restrict ourselves with simplest case when the dressing factor has a single pair of poles. We shall also assume that and $X$ and $F$ are 3-vectors $X^T = (X^1, X^2, X^3)$ and $F^T = (F^1, F^2, F^3)$ respectively. So (\[f\_psi0\]) leads to the following expression for $F$: $$F(x) = \left(\begin{array}{c}
F_{0}^{1}\cos\mu x + \rmi F_{0}^{3}\sin \mu x \\
F_{0}^{2} \\ F_{0}^{3}\cos \mu x + \rmi F_{0}^{1}\sin \mu x 
\end{array}\right),\qquad F_0   = \left(\begin{array}{c}
F_{0}^{1} \\ F_{0}^{2} \\ F_{0}^{3} \end{array}\right).
\label{f_sol}$$ In order to evaluate $X$, we shall need the following quadratic expressions as well: $$\begin{aligned}
F^T(x)QF^*(-x) & = & \left(\left |F_{0}^{1}\right |^2 + |F_{0}^{3}|^2\right)\cos{2\omega x}
+ 2\rmi |F_{0}^{1}F_{0}^{3}|\cos{2\varphi}\sin{2\omega x} + \varepsilon |F_{0}^{2}|^2,\label{ff1}\\
F^T(x)QHF^*(-x) & = & \left[\left(|F_{0}^{3}|^2 - |F_{0}^{1}|^2\right)\cosh{2\kappa x} 
- 2\rmi |F_{0}^{1}F_{0}^{3}|\sin{2\varphi}\sinh{2\kappa x} + \varepsilon|F_{0}^{2}|^2 \right]\label{ff2}\end{aligned}$$ where $\omega=\re\mu>0$, $\kappa=\im\mu>0$ and $2\varphi = \arg F_{0}^{1} - \arg F_{0}^{3}$. Taking into account the structure of (\[psi\_0\]) and (\[f\_sol\]), it is natural to consider in detail the following cases, see [@side9; @yantih2].

First, let us assume $F_{0}^{2} = 0$ and $F_{0}^{1}\neq \pm F_{0}^{3}$. Then for $F$ we have $$F(x) = \left(\begin{array}{c}
F_{0}^{1}\cos\mu x  + \rmi F_{0}^{3}\sin \mu x \\
0 \\ F_{0}^{3}\cos \mu x + \rmi F_{0}^{1}\sin \mu x 
\end{array}\right).
\label{f_sol1}$$ After substituting (\[f\_sol1\]) into (\[XF\_sys\]) and (\[s10\_eq\]), we get the dressed solution at a fixed moment of time: $$\begin{aligned}
u_1(x) & = & 0\, , \label{u1_1}\\
v_1(x) & = & \left[\frac{c_1\cosh2\kappa x - c_2\sin2\omega x
+ \rmi (c_3\cos 2\omega x + c_4\sinh 2\kappa x)}{c_1\cosh2\kappa x
+ c_2\sin2\omega x - \rmi (c_3\cos 2\omega x - c_4\sinh 2\kappa x)}\right]^2 .
\label{v1_1}\end{aligned}$$ where $2\gamma = \ln|F^1_{0}/F^3_{0}|$ and $$c_1 := \omega\sinh2\gamma,\qquad c_2 := \kappa\cos2\varphi,\qquad c_3 : = \kappa\cosh2\gamma,
\qquad c_4 := \omega\sin2\varphi .
\label{c1234}$$ To recover the time evolution in (\[u1\_1\]) and (\[v1\_1\]), one needs to make the substitution that follows directly from (\[f\_j0\_evol\]). It is seen that (\[u1\_1\]) and (\[v1\_1\]) remain invariant under that transformation so the dressed solution is stationary.

Let us assume $F_{0}^{2} \neq 0$ . Without any loss of generality we could set $F_{0}^{2} = 1$. After recovering the time evolution, the vector $F(x)$ is given by: $$F(x,t) = \left(\begin{array}{c}
 \rme^{\frac{\rmi \mu^2 t}{3}}\left(F_{0}^{1}\cos\mu x + \rmi F^3_{0}\sin\mu x\right) \\
\rme^{-\frac{2\rmi \mu^2 t}{3}} \\ \rme^{\frac{\rmi \mu^2 t}{3}}\left(F_{0}^3\cos\mu x + \rmi F_{0}^1\sin\mu x\right) 
\end{array}\right).$$ The solution in this case reads: $$\begin{aligned}
u_1 = \frac{\left[c_1\cosh2\kappa x - c_2\sin2\omega x
+ \rmi (c_3\cos 2\omega x + c_4\sinh 2\kappa x) - \varepsilon(\omega - \rmi \kappa)\rme^{2(2\omega\kappa t - \xi_0)}/2\right]}{\left[c_1\cosh2\kappa x
+ c_2\sin2\omega x - \rmi (c_3\cos 2\omega x - c_4\sinh 2\kappa x)
- \varepsilon(\omega + \rmi\kappa)\rme^{2(2\omega\kappa t - \xi_0)}/2\right]^2} \nonumber\\
\times \frac{2\rmi\omega\kappa}{\omega- \rmi \kappa}\rme^{-\rmi[(\omega^2 - \kappa^2)t + \delta_0]}
\left[\rme^{\rmi\varphi - \gamma}\cos(\omega + \rmi \kappa)x + \rmi \rme^{-\rmi\varphi + \gamma}
\sin(\omega + \rmi \kappa)x\right]\rme^{2\omega\kappa t - \xi_0}
\label{u1_2}\\
v_1 = \frac{\left[c_1\cosh2\kappa x - c_2\sin2\omega x
+ \rmi (c_3\cos 2\omega x + c_4\sinh 2\kappa x) - \varepsilon(\omega - \rmi \kappa)\rme^{2(2\omega\kappa t - \xi_0)}/2\right]}{\left[c_1\cosh2\kappa x
+ c_2\sin2\omega x - \rmi (c_3\cos 2\omega x - c_4\sinh 2\kappa x)
- \varepsilon(\omega + \rmi\kappa)\rme^{2(2\omega\kappa t - \xi_0)}/2\right]^2}\nonumber\\
\times \frac{\left\{\left[c_1\cosh2\kappa x - c_2\sin2\omega x
+ \rmi (c_3\cos 2\omega x + c_4\sinh 2\kappa x) - \varepsilon(\omega - \rmi \kappa)\rme^{2(2\omega\kappa t - \xi_0)}/2\right]\right.}
{\omega - \rmi\kappa} \\
\left.\times (\omega - \rmi \kappa) - 2\rmi\omega\kappa\varepsilon\rme^{2(2\omega\kappa t - \xi_0)}\right\}\nonumber
\label{v1_2}\end{aligned}$$ where the coefficients $c_j$, $j=1,2,3,4$ are the same as in (\[c1234\]) and $$2\gamma = \ln|F^1_{0}/F^3_{0}|, \quad 2\varphi = \arg F_{0}^{1} - \arg F_{0}^{3},\quad
2\delta_0 = \arg F_{0}^{1} + \arg F_{0}^{3},\quad 2\xi_0 = \ln|F_0^1F_0^3|$$

All the solutions constructed explicitly in here are connected with four distinct poles of $\mathcal{G}$ and $\hat{\mathcal{G}}$, i.e. four discrete eigenvalues of the dressed scattering operator.

Integrals of Motion {#consdens}
===================

Condition (\[constr\]) implies that $S$ has a simple spectrum — its eigenvalues are $0$, $\pm 1$. Thus we can put $S(x,t)$ into a constant diagonal form by applying gauge transformation $$\begin{aligned}
L(\lambda) &\to & \tilde{L}(\lambda) = \hat{g}(x,t)L(\lambda)g(x,t)
= \rmi \partial_x + \tilde{U}_0(x,t) - \lambda J,\quad J  = \diag(1,0,-1) \label{l_tild}\\
A(\lambda) &\to & \tilde{A}(\lambda) = \hat{g}(x,t) A(\lambda)g(x,t) 
= \rmi\partial_t + \sum^2_{k=0}\tilde{A}_k(x,t)\lambda^k,\label{m_tild}\\
\hat{g} & := & g^{-1},
\quad g(x,t) = \frac{\sqrt{2}}{2}\left(\begin{array}{ccc}
1 & 0 & -1 \\ \varepsilon u^*(-x,t) & \sqrt{2} v(x,t) & \varepsilon u^*(-x,t)\\
v^*(-x,t) & -\sqrt{2} u(x,t)  & v^*(-x,t)
\end{array}\right)
\label{g_trans}\end{aligned}$$ on the Lax operators. Further on we are also going to use the explicit form of $\tilde{U}_0$ given by $$\tilde{U}_0(x,t)  = \frac{\rmi}{2}\left(\begin{array}{ccc}
B(x,t) & C(x,t) & B(x,t) \\
-\varepsilon C^*(-x,t) & -2B(x,t) 
& -\varepsilon C^*(-x,t)\\
B(x,t) & C(x,t) & B(x,t)
\end{array}\right)
\label{u_0}$$ where $C(x,t) = \sqrt{2}[u(x,t)v_x(x,t) - v(x,t)u_x(x,t)]$.

Let us apply Drinfel’d and Sokolov’s method of diagonalization of Lax pair [@DrSok*85] to derive the integrals of motion of (\[ghf\_nonloc\]). Introduce the gauge transform $$\mathcal{P}(x,t,\lambda)= \openone +\frac{P_1(x,t)}{\lambda} +
\frac{P_2(x,t)}{\lambda^2} + \cdots \label{P_trans}$$ where all coefficients $P_l$ ($l=1,2,\ldots$) are off-diagonal matrices to be determined further. As a result we have $$\begin{aligned}
\mathcal{L} & = & \hat{\mathcal{P}}\tilde{L}\mathcal{P}
= \rmi \partial_x - \lambda J + \mathcal{L}_0
+ \frac{\mathcal{L}_1}{\lambda} + \cdots,\label{L_1}\\
\mathcal{A} & = & \hat{\mathcal{P}}\tilde{A}\mathcal{P}
= \rmi \partial_t + \sum^{\infty}_{k=-2}\lambda^{-k} \mathcal{A}_{k}.
\label{Mu}\end{aligned}$$ All coefficients $\mathcal{L}_k$, $\mathcal{A}_{k}$, $k=-1,0,1,\hdots$ are required to be diagonal matrices. Then the zero curvature representation $[\mathcal{L}, \mathcal{A}] = 0$ is equivalent to the following equations $$\begin{gathered}
\partial_t\mathcal{L}_k - \partial_x \mathcal{A}_k =  0,\qquad
k=0,1,\dots.\end{gathered}$$ These equations mean that $\mathcal{L}_k$ are conserved densities of (\[ghf\_nonloc\]) and $\mathcal{A}_k$ are the corresponding currents.

Let us rewrite equality (\[Lambda\]) as follows: $$\tilde{L}\mathcal{P} = \mathcal{P}\mathcal{L}
\label{lppl}$$ Since (\[lppl\]) should hold identically in $\lambda$, it splits into the recurrence relations: $$\begin{aligned}
\lambda^{0}  &:& \tilde{U}_0 - J P_1 = \mathcal{L}_0 - P_1 J,\\
\lambda^{-1} &:& \rmi P_{1,x} + \tilde{U}_0 P_1 - JP_2
= \mathcal{L}_1 + P_1\mathcal{L}_0 - P_2 J, \label{lambda_m1}\\
& & \ldots \nonumber\\
\lambda^{-k} &:& \rmi P_{k,x} + \tilde{U}_0 P_k - JP_{k+1}
= \mathcal{L}_k - P_{k+1}J + \sum^{k-1}_{m = 0}P_{k-m}\mathcal{L}_m,
\label{lambda_mk}\\ 
&&\cdots \nonumber\end{aligned}$$ In order to solve those, one should split each relation into a diagonal and off-diagonal part. For example, from the first relation above one has $$\mathcal{L}_0 = \tilde{U}^{\rm d}_0,\qquad \tilde{U}^{\rm a}_0 =  [J,P_1],
\label{recursplit_1}$$ where the superscripts $\rm d$ and $\rm a$ above denote projection onto diagonal and off-diagonal part of a matrix respectively. Taking into account the explicit form of $\tilde{U}_0$ (formula (\[u\_0\])) for $\mathcal{L}_0$ we have $$\mathcal{L}_0 = \frac{\rmi}{2}B(x,t)\left(\begin{array}{ccc}
1 & 0  & 0 \\
0 & -2 & 0 \\
0 & 0  & 1
\end{array}\right).$$ Thus as a density of our first integral we can choose: $\mathcal{I}_1 = - B(x,t)$. It represents momentum density of our system.

After inverting the commutator in the second equation in (\[recursplit\_1\]) one obtains $$P_1 = \frac{\rmi}{2}\left(\begin{array}{ccc}
0  & C(x,t) & B(x,t)/2 \\
\varepsilon C^*(-x,t) & 0 & -\varepsilon C^*(-x,t)\\
-B(x,t)/2 & -C(x,t) & 0
\end{array}\right).
\label{t_1}$$ Similarly, for $\mathcal{L}_1$ one needs to extract the diagonal part of (\[lambda\_m1\]). The result reads $$\begin{gathered}
\mathcal{L}_1 = \left(\tilde{U}^{\rm a}_0 P_1\right)^{\rm d}.
\label{recursplit_2d}\end{gathered}$$ After substituting the expression (\[t\_1\]) for $P_1$ into (\[recursplit\_2d\]) one obtains $$\mathcal{L}_1 = \frac{1}{8}\left[B(x,t)^2
- 4\varepsilon C(x,t)C^*(-x,t)\right]
\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -1
\end{array}\right).$$ Hence as a second integral density one can pick up $$\mathcal{I}_2 = B(x,t)^2 -2\varepsilon C(x,t)C^*(-x,t).$$

Proceeding that way one is able to find conserved density of arbitrary order. It is seen from (\[lambda\_mk\]) that $k$-th coefficient $\mathcal{L}_k$ is obtained from formula $$\mathcal{L}_k = \big(\tilde{U}^{\rm a}_0P_k\big)^{\rm d}.
\label{l_k}$$ while $P_k$ is recursively obtained from: $$P_k = \ad^{-1}_{J}\left(\rmi \partial_x P_{k-1}
+ (\tilde{U}_0 P_{k-1})^{\rm a} - \sum^{k-2}_{l=0}P_{k-1-l}\mathcal{L}_l\right).$$

In order to obtain an integral of motion of (\[ghf\_nonloc\]), one needs to integrate $$I_k = \int^{\infty}_{-\infty}\rmd x \mathcal{I}_k, \qquad k=1,2,\ldots$$

Conclusion
==========

In the present paper we have studied a nonlocal reduction of a generalized Heisenberg ferromagnet equation, see (\[ghf\_nonloc\]). We have described the integrable hierarchy of (\[ghf\_nonloc\]) by using $\Lambda$ operators, see (\[int\_hier\]). Moreover, using rational dressing technique, we have constructed particular solutions to (\[ghf\_nonloc\]). Similarly one can derive solutions to any member of the same integrable hierarchy. The only difference will be the dispersion law, hence the time dependence of the solution. We have considered here particular solutions of quadruplet type, i.e. those related to 4 complex discrete eigenvalues of the scattering operator. There are other types of solutions like doublet solutions and quasi-rational solutions. Constructing the latter two types of solutions will be done elsewhere.

We have also paid certain attention to conserved densities for the system of nonlocal equations. By implementing Drinfel’d-Sokolov’s method, we have derived recursive formulas to obtain such densities, see (\[l\_k\]).

Acknowledgments {#acknowledgments .unnumbered}
===============

The work has been supported by grant DN 02-5 of Bulgarian Fund “Scientific Research”.

[00]{} Borovik A. E. and Popkov V. Yu., Completely Integrable Spin - 1 Chains, [*Sov. Phys. JETPH*]{} [**71**]{} (1990) 177–85. Drinfel’d V. G., Sokolov V.V., Lie algebras and equations of Korteweg–de Vries type, [*J. Math. Sci.*]{} [**30**]{} (1985), 1975–2036. Gerdjikov V., Vilasi G. and Yanovski A., Integrable Hamiltonian Hierarchies, Springer: Berlin, 2008. Gerdjikov V. S., Mikhailov A. V. and Valchev T. I., Reductions of Integrable Equations on A.III-type Symmetric Spaces, [*J. Phys. A: Math. Theor.*]{} [**43**]{} (2010) 434015. Gerdjikov V. S., Mikhailov A. V. and Valchev T. I., Recursion Operators and Reductions of Integrable Equations on Symmetric Spaces, [*J. Geom. and Symm. in Phys.*]{} [**20**]{} (2010) 1–34. Gerdjikov V., Grahovski G., Mikhailov A. and Valchev T., Polynomial Bundles and Generalized Fourier Transforms for Integrable Equations on [**A.III**]{}-type Symmetric Spaces, *SIGMA* **7**, 096 (2011) 48 pages. Mikhailov A. V., Reduction in the Integrable Systems. Reduction Groups, [*Lett. JETF*]{} [**32**]{} (1979) 187–192. Mikhailov A. V., The Reduction Problem and Inverse Scattering Method, [*Physica D*]{} [**3**]{} (1981) 73–117. Takhtadjan L. and Faddeev L., The Hamiltonian Approach to Soliton Theory, Springer Verlag, Berlin, (1987). Yanovski A. B. and Valchev T. I., Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. I. Auxiliary System and Fundamental Properties, [*J. Nonl. Math. Phys.*]{} [**25**]{} (2018) ArXiv: 1709.09266 \[nlin.SI\]. Valchev T. and Yanovski A., Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. II. Special Solutions, [*J. Nonl. Math. Phys.*]{} [**25**]{} (2018) 442–461, ArXiv 1711.06353v2. Yanovski A. B., On the Recursion Operators for the Gerdjikov, Mikhailov and Valchev System, [*J. Math. Phys.*]{} **52**, 8 (2011) 082703. Yanovski A. and Vilasi G., Geometry of the Recursion Operators for the GMV System, [*J. Nonl. Math. Phys.*]{} **19** N3 (2012) 1250023-1/18. Zakharov V. and Mikhailov A., On the Integrability of Classical Spinor Models in Two-Dimensional Space-Time, [*Commun. Math. Phys.*]{} [**74**]{} (1980) 21–40. Zakharov V. E. and Shabat A. B., A Scheme for Integrating Nonlinear Equations of Mathematical Physics by the Method of the Inverse Scattering Transform II, [*Funct. Anal. and Appl.*]{} [**13**]{} (1979) 13–23.
