ODE/IM correspondence for the Fateev model

The Fateev model is somewhat special among two-dimensional quantum field theories. For different values of the parameters,it can be reduced to a variety of integrable systems. An incomplete list of the reductions includes O(3) and O(4) non-linear sigma models and their continuous deformations (2D and 3D sausages, anisotropic principal chiral field), the Bukhvostov-Lipatov model, the N=2 supersymmetric sine-Gordon model, as well as the integrable perturbed SU_2(n)\otimes SU_2(p-2)/SU_2(n+p-2) coset CFT. The model possesses a mysterious symmetry structure of the exceptional quantum superalgebras U_q{\hat{\big(D(2|1;\alpha)}}\big). In this work, we propose the ODE/IM correspondence between the Fateev model and a certain generalization of the classical problem of constant mean curvature embedding of a thrice-punctured sphere in AdS_3.