We establish an intriguing correspondence between a special set of classical solutions of the modified sinh-Gordon equation (i.e., Hitchin's ‘self-duality’ equations) on a punctured Riemann sphere and a set of stationary states in the finite-volume Hilbert space of the integrable 2D quantum field theory introduced by VA Fateev. An application of this correspondence to the problem of non-perturbative quantization of classically integrable nonlinear sigma models is briefly discussed. A detailed account of the results announced in this communication is contained in separate publications (Bazhanov and Lukyanov 2014 arXiv:1310.4390 [hep-th] and Bazhanov et al 2014 J. High Energy Phys. JHEP09(2014)147).