Interacting diffusive particle systems are paradigms for non-equilibrium statistical physics. Their macroscopic behaviour follows a variational principle, proposed by G. Jona-Lasinio and his collaborators, known as the Macroscopic Fluctuation Theory (MFT), in which physics out from equilibrium is determined at a coarse-grained scale by two coupled non-linear hydrodynamic equations. In this talk, we shall show that the MFT equations for the exclusion process are classically integrable, i.e. they can be integrated by inverse scattering, a method originally used to study solitons in the KdV or the NLS equations. Our exact solution will allow us to understand how large deviations are generated by atypical fluctuations, far from equilibrium.