Analytic (non)integrability of Arutyunov-Bassi-Lacroix model

We use the notion of the gauge/string duality and discuss the Liouvillian (non) integrability criteria for string sigma models in the context of recently proposed Arutyunov-Bassi-Lacroix (ABL) model [JHEP $\mathbf{\text{03}}$ (2021), 062]. Our analysis complements those previous results due to numerical analysis as well as Lax pair formulation. We consider a winding string ansatz for the deformed torus $T_k^{\text{qty}({\lambda }_1,{\lambda }_2,\lambda )}$ which can be interpreted as a system of coupled pendulums. Our analysis reveals the Liouvillian nonintegrablity of the associated sigma model. We also obtain the \emph{generalized} decoupling limit and confirm the analytic integrability for the decoupled sector.