In this chapter we study the symmetric harmonic process, and its continuous equivalent, the integrable heat conduction model. These are processes with \(\mathfrak {su}(1,1)\) symmetry which in a setting of a chain with left and right reservoirs are exactly solvable, i.e., we have close-form expressions for the moments in the non-equilibrium steady state and obtain an explicit expression of the stationary state as a mixture of product measures. We introduce both models, their algebraic structure, their dualities, and the connection with an integrable quantum spin chain with non-compact spins. The duality functions coincide with those of, respectively, the inclusion process (Chap. 5 ) and the Brownian energy process (Chap. 6 ), as they share the same underlying Lie algebra and the same symmetries. We introduce the so-called hidden parameter model which describes the evolution of product measures with equilibrium marginals. We show the Markovian structure of the hidden parameter model and provide a probabilistic proof of the structure of the non-equilibrium steady state of the model on a chain with left and right reservoirs. We also discuss an explicit formula for the rate function describing the large deviations of the density profile in the non-equilibrium steady state.