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1. Geometry of Integrable Linkages

   https://doi.org/10.56994/JXM.001.001.005  

   Perline, Ron; Tabachnikov, Sergei (February 2025, Journal of Experimental Mathematics)

   In analogy with the well-known 2-linkage tractor-trailer problem, we define a 2-linkage problem in the plane with novel non-holonomic “no-slip” conditions. Using constructs from sub-Riemannian geometry, we look for geodesics corresponding to linkage motion with these constraints (“tricycle kinematics”). The paths of the three vertices turn out to be critical points for functionals which appear in the hierarchy of conserved quantities for the planar filament equation, a well known completely integrable evolution equation for planar curves. We show that the geodesic equations are completely integrable, and present a second connection to the planar filament equation. 

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2. On the Lie algebra structure of integrable derivations

   https://doi.org/10.1112/blms.12884  

   Briggs, Benjamin; Rubio_y_Degrassi, Lleonard (December 2023, Bulletin of the London Mathematical Society)

   Abstract Building on work of Gerstenhaber, we show that the space of integrable derivations on an Artin algebra forms a Lie algebra, and a restricted Lie algebra if contains a field of characteristic . We deduce that the space of integrable classes in forms a (restricted) Lie algebra that is invariant under derived equivalences, and under stable equivalences of Morita type between self‐injective algebras. We also provide negative answers to questions about integrable derivations posed by Linckelmann and by Farkas, Geiss and Marcos. Along the way, we compute the first Hochschild cohomology of the group algebra of any symmetric group. 

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3. Invariant Measures for Integrable Spin Chains and an Integrable Discrete Nonlinear Schrödinger Equation

   https://doi.org/10.1137/19M1265314  

   Angelopoulos, Yannis; Killip, Rowan; Visan, Monica (January 2020, SIAM Journal on Mathematical Analysis)
4. Fractional Integrable Nonlinear Soliton Equations

   https://doi.org/10.1103/PhysRevLett.128.184101  

   Ablowitz, Mark J.; Been, Joel B.; Carr, Lincoln D. (May 2022, Physical Review Letters)
5. On polynomially integrable convex bodies

   Koldobsky, Alexander; Merkurjev, Alexander; Yaskin, Vladyslav (October 2017, Advances in mathematics)