{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:36:13Z","timestamp":1760229373800,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,9]],"date-time":"2022-06-09T00:00:00Z","timestamp":1654732800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11871336","11771395","11975145","11972291"],"award-info":[{"award-number":["11871336","11771395","11975145","11972291"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The algebraic structures of zero curvature representations are furnished for multilayer integrable couplings associated with matrix spectral problems, both discrete and continuous. The key elements are a class of matrix loop algebras consisting of block matrices with blocks of the same size. As illustrative examples, isospectral and non-isospectral integrable couplings and the corresponding commutator relations of their Lax operators are computed explicitly in the cases of the Volterra lattice hierarchy and the AKNS hierarchy, along with their \u03c4-symmetry algebras.<\/jats:p>","DOI":"10.3390\/sym14061185","type":"journal-article","created":{"date-parts":[[2022,6,13]],"date-time":"2022-06-13T02:01:44Z","timestamp":1655085704000},"page":"1185","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Lax Operator Algebras and Applications to \u03c4-Symmetries for Multilayer Integrable Couplings"],"prefix":"10.3390","volume":"14","author":[{"given":"Chun-Xia","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Capital Normal University, Beijing 100048, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5309-1493","authenticated-orcid":false,"given":"Wen-Xiu","family":"Ma","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China"},{"name":"Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2573-0686","authenticated-orcid":false,"given":"Shou-Feng","family":"Shen","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1002\/sapm1974534249","article-title":"The inverse scattering transform-Fourier analysis for nonlinear problems","volume":"53","author":"Ablowitz","year":"1974","journal-title":"Stud. 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