2026/06/24 by Alexander V. Mikhailov, Vladimir Novikov, Jing Ping Wang · 1 voice
Physics and Astronomy · #nlin.SI
arxiv published 2026/06/24 · arxiv updated 2026/06/27
We derive necessary and sufficient integrability conditions for two-component polynomial evolutionary systems of odd order in (1+1) dimensions. Integrable systems are members of infinite hierarchies of commuting symmetries, which are characterised by their spectral invariants. We prove that there are precisely 24 possible spectral classes of integrable hierarchies. As an application, we obtain a complete classification of integrable homogeneous hierarchies whose lowest-order equations are of order 3 and 5. The resulting classification naturally splits into two classes. The C--integrable systems are reduced, by means of differential substitutions, to linear--triangular form, while the S--integrable systems are related, through linear changes of variables and differential substitutions, to canonical Drinfeld--Sokolov KdV-type systems associated with affine Lie algebras of rank two.