{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T18:35:13Z","timestamp":1766428513330,"version":"build-2065373602"},"reference-count":65,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,15]],"date-time":"2025-01-15T00:00:00Z","timestamp":1736899200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We study an interesting superization problem of integrable nonlinear dynamical systems on functional manifolds. As an example, we considered a quantum many-particle Schr\u00f6dinger\u2013Davydov model on the axis, whose quasi-classical reduction proved to be a completely integrable Hamiltonian system on a smooth functional manifold. We checked that the so-called \u201cnaive\u201d approach, based on the superization of the related phase space variables via extending the corresponding Poisson brackets upon the related functional supermanifold, fails to retain the dynamical system super-integrability. Moreover, we demonstrated that there exists a wide class of classical Lax-type integrable nonlinear dynamical systems on axes in relation to which a superization scheme consists in a reasonable superization of the related Lax-type representation by means of passing from the basic algebra of pseudo-differential operators on the axis to the corresponding superalgebra of super-pseudodifferential operators on the superaxis.<\/jats:p>","DOI":"10.3390\/sym17010125","type":"journal-article","created":{"date-parts":[[2025,1,15]],"date-time":"2025-01-15T10:00:01Z","timestamp":1736935201000},"page":"125","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On Superization of Nonlinear Integrable Dynamical Systems"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5124-5890","authenticated-orcid":false,"given":"Anatolij K.","family":"Prykarpatski","sequence":"first","affiliation":[{"name":"Department of Computer Science and Telecommunication, Cracow University of Technology, 31-155 Krak\u00f3w, Poland"},{"name":"Department of Advanced Mathematics, Lviv Polytechnic National University, 79013 Lviv, Ukraine"}]},{"given":"Rados\u0142aw A.","family":"Kycia","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Telecommunication, Cracow University of Technology, 31-155 Krak\u00f3w, Poland"}]},{"given":"Volodymyr M.","family":"Dilnyi","sequence":"additional","affiliation":[{"name":"Department of Advanced Mathematics, Lviv Polytechnic National University, 79013 Lviv, Ukraine"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/0370-2693(85)91326-7","article-title":"Infinite family of polynomial functions of the Virassoro generators with vanishing Poisson brackets","volume":"160","author":"Gervais","year":"1985","journal-title":"Phy. 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