We evaluate the full statistics of non-stationary heat transfer in the one-dimensional Kipnis- Marchioro-Presutti lattice gas model at long times by exploiting complete integrability of the equations of the macroscopic fluctuation theory for this model. These equations are closely related to the derivative nonlinear Schr odinger equation (DNLS), and we solve them by the Zakharov-Shabat inverse scattering method (ISM) adapted by Kaup and Newell (1978) for the DNLS.