  We determine the full statistics of nonstationary heat transfer in the
Kipnis-Marchioro-Presutti lattice gas model at long times by uncovering and
exploiting complete integrability of the underlying equations of the
macroscopic fluctuation theory. These equations are closely related to the
derivative nonlinear Schr\"{o}dinger equation (DNLS), and we solve them by the
Zakharov-Shabat inverse scattering method (ISM) adapted by Kaup and Newell
(1978) for the DNLS. We obtain explicit results for the exact large deviation
function of the transferred heat for an initially localized heat pulse, where
we uncover a nontrivial symmetry relation.

Statistical Mechanics;Probability