# Integrable (2 + 1)-Dimensional Spin Models with Self-Consistent Potentials

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## Abstract

**:**

## 1. Introduction

## 2. A Brief Review on Integrable Spin Systems in 2 + 1 Dimensions

#### 2.1. The Ishimori Equation

#### 2.2. The Myrzakulov-I Equation

#### 2.3. The Myrzakulov–Lakshmanan I Equation

#### 2.4. The (2 + 1)-Dimensional Heisenberg Ferromagnet Equation

## 3. The Myrzakulov–Lakshmanan II Equation

#### 3.1. Reductions

#### 3.2. Lax Representation

#### 3.3. Gauge Equivalent Counterpart of the ML-II Equation

#### 3.4. Integral of Motion

## 4. The Myrzakulov–Lakshmanan III Equation

#### 4.1. Lax Representation

#### 4.2. Reductions

#### 4.2.1. Case I: ${\u03f5}_{2}=0$

#### 4.2.2. Case II: $W=0$

#### 4.2.3. Case III: $y=x$

#### 4.3. Equivalent Counterpart of the ML-III Equation

## 5. The Myrzakulov–Lakshmanan IV Equation

#### 5.1. Lax Representation

#### 5.2. Reductions

#### 5.2.1. Case I: ${\u03f5}_{1}={\u03f5}_{2}=0$

#### 5.2.2. Case II: ${\u03f5}_{1}\ne 0,{\u03f5}_{2}=0$

#### 5.2.3. Case III: ${\u03f5}_{1}=0,{\u03f5}_{2}\ne 0$

#### 5.2.4. Case IV: $W=0$

#### 5.2.5. Case V: $y=x$

#### 5.3. Equivalent Counterpart of the ML-IV Equation

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A: Gauge Equivalence

## Appendix B: Nonisospectral Problem

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Myrzakulov, R.; Mamyrbekova, G.; Nugmanova, G.; Lakshmanan, M.
Integrable (2 + 1)-Dimensional Spin Models with Self-Consistent Potentials. *Symmetry* **2015**, *7*, 1352-1375.
https://doi.org/10.3390/sym7031352

**AMA Style**

Myrzakulov R, Mamyrbekova G, Nugmanova G, Lakshmanan M.
Integrable (2 + 1)-Dimensional Spin Models with Self-Consistent Potentials. *Symmetry*. 2015; 7(3):1352-1375.
https://doi.org/10.3390/sym7031352

**Chicago/Turabian Style**

Myrzakulov, Ratbay, Galya Mamyrbekova, Gulgassyl Nugmanova, and Muthusamy Lakshmanan.
2015. "Integrable (2 + 1)-Dimensional Spin Models with Self-Consistent Potentials" *Symmetry* 7, no. 3: 1352-1375.
https://doi.org/10.3390/sym7031352