{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T01:35:34Z","timestamp":1772760934737,"version":"3.50.1"},"reference-count":46,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,2]],"date-time":"2025-07-02T00:00:00Z","timestamp":1751414400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan","award":["AP25794905"],"award-info":[{"award-number":["AP25794905"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper investigates an integrable spin system known as the Myrzakulov-XIII (M-XIII) equation through geometric and gauge-theoretic methods. The M-XIII equation, which describes dispersionless dynamics with curvature-induced interactions, is shown to admit a geometric interpretation via curve flows in three-dimensional space. We establish its gauge equivalence with the complex coupled dispersionless (CCD) system and construct the corresponding Lax pair. Using the Sym\u2013Tafel formula, we derive exact soliton surfaces associated with the integrable evolution of curves and surfaces. A key focus is placed on the role of geometric and gauge symmetry in the integrability structure and solution construction. The main contributions of this work include: (i) a commutative diagram illustrating the connections between the M-XIII, CCD, and surface deformation models; (ii) the derivation of new exact solutions for a fractional extension of the M-XIII equation using the Kudryashov method; and (iii) the classification of these solutions into trigonometric, hyperbolic, and exponential types. These findings deepen the interplay between symmetry, geometry, and soliton theory in nonlinear spin systems.<\/jats:p>","DOI":"10.3390\/sym17071041","type":"journal-article","created":{"date-parts":[[2025,7,2]],"date-time":"2025-07-02T06:10:26Z","timestamp":1751436626000},"page":"1041","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Ferromagnet-Type System: Integrable Flows of Curves\/Surfaces, Soliton Solutions, and Equivalence"],"prefix":"10.3390","volume":"17","author":[{"given":"Gulgassyl","family":"Nugmanova","sequence":"first","affiliation":[{"name":"Department of Mathematical and Computer Modeling, L.N. Gumilyov Eurasian National University, Astana 010000, Kazakhstan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7310-1185","authenticated-orcid":false,"given":"Guldana","family":"Bekova","sequence":"additional","affiliation":[{"name":"Department of General and Theoretical Physics, L.N. Gumilyov Eurasian National University, Astana 010000, Kazakhstan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8491-0893","authenticated-orcid":false,"given":"Meruyert","family":"Zhassybayeva","sequence":"additional","affiliation":[{"name":"Department of General and Theoretical Physics, L.N. Gumilyov Eurasian National University, Astana 010000, Kazakhstan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5200-0393","authenticated-orcid":false,"given":"Aigul","family":"Taishiyeva","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Methods of Teaching Mathematics, Kh. Dosmukhamedov Atyrau University, Atyrau 060000, Kazakhstan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7351-0472","authenticated-orcid":false,"given":"Kuralay","family":"Yesmakhanova","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Computer Modeling, L.N. Gumilyov Eurasian National University, Astana 010000, Kazakhstan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4047-4484","authenticated-orcid":false,"given":"Zhaidary","family":"Myrzakulova","sequence":"additional","affiliation":[{"name":"Department of Algebra and Geometry, L.N. Gumilyov Eurasian National University, Astana 010000, Kazakhstan"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,2]]},"reference":[{"key":"ref_1","first-page":"62","article-title":"Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media","volume":"34","author":"Zakharov","year":"1972","journal-title":"Sov. Phys. JETP"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1280","DOI":"10.1098\/rsta.2010.0319","article-title":"The Fascinating World of the Landau-Lifshitz-Gilbert Equation: An Overview","volume":"369","author":"Lakshmanan","year":"2011","journal-title":"Phil. Trans. R. Soc. 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