  We establish a correspondence between an infinite set of special solutions of
the (classical) modified sinh-Gordon equation and a set of stationary states in
the finite-volume Hilbert space of the integrable 2D QFT invented by V.A.
Fateev. The modified sinh-Gordon equation arise in this case as a
zero-curvature condition for a class of multivalued connections of the
punctured Riemann sphere, similarly to Hitchin's self-duality equations. The
proposed correspondence between the classical and quantum integrable systems
provides a powerful tool for deriving functional and integral equations which
determine the full spectrum of local integrals of motion for massive QFT in a
finite volume. Potential applications of our results to the problem of
non-perturbative quantization of classically integrable non-linear sigma models
are briefly discussed.
