Integrable Zhaidary equations: reductions and gauge equivalence

TL;DR

The paper investigates the integrability of the Zhaidary equation (ZE), a 1+1D spin system for a unit vector ${\bf S}$ with scalar potentials, by constructing a gauge partner (Zhaidary-II) and deriving its integrable reductions (Kuralay and Shynaray) via explicit Lax pairs. It then extends the ZE framework with several integrable generalizations (Nurshuak, Aizhan, Zhanbota, Akbota families) and their gauge equivalents, broadening the integrable spin-system landscape. In the 1+0D setting, Zhanbota transcendents are connected to the six Painlevé equations through gauge equivalences for Zhanbota-III to Zhanbota-IV, linking spin dynamics to classical Painlevé theory. The work assembles explicit Lax representations, gauge maps, and reductions to reveal a cohesive network between ZE, its generalizations, and Painlevé equations, with implications for geometry and potential discrete/nonlocal extensions.

Abstract

The present work addresses the study and characterization of the integrability of some generalized spin systems (ISS) in 1+1 dimensions. Lax representations for these ISS are successfully obtained. The gauge equivalent counterparts of these integrable ISS are presented. Finally, we consider Zhanbota transcendents and some integrable Zhanbota equations. In particular, the gauge equivalence between some Zhanbota equations and the six Painleve equations is established.