  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.
