Geodesic
Invariants of homogeneous dynamic systems of arbitrary odd order with dissipation. IV. Ninth-order systems
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the 6th International Conference "Dynamic Systems and Computer Science: Theory and Applications" (DYSC 2024). Irkutsk, September 16-20, 2024. Part 2, Tome 239 (2025), pp. 62-97

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In this paper, we present new examples of integrable dynamical systems of the ninth order that are homogeneous in part of the variables. In these systems, subsystems on the tangent bundles of lower-dimensional manifolds can be distinguished. In the cases considered, the force field is partitioned into an internal (conservative) part and an external part. The external force introduced by a certain unimodular transformation has alternate dissipation; it is a generalization of fields examined earlier. Complete sets of first integrals and invariant differential forms are presented. The first part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 236 (2024), pp. 72–88. The second part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 237 (2024), pp. 49–75. The third part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 238 (2024), pp. 69–100.
Keywords: dynamical system, integrability, dissipation, first integral with essential singular points, invariant differential form 
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M. V. Shamolin. Invariants of homogeneous dynamic systems of arbitrary odd order with dissipation. IV. Ninth-order systems. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the 6th International Conference "Dynamic Systems and Computer Science: Theory and Applications" (DYSC 2024). Irkutsk, September 16-20, 2024. Part 2, Tome 239 (2025), pp. 62-97. http://geodesic.mathdoc.fr/item/INTO_2025_239_a6/