We consider a class of large-scale interacting systems with one conservation law satisfying the ``degree-preserving property'', and study the classification of their invariant measures and their hydrodynamic limits. Under a few basic conditions, we show that if the generator of the process preserves the degree of polynomials of the state variables up to two, then the marginals of any product invariant measure of the process must belong to one of six specific distributions. This classification result is essentially a consequence of a known result in statistics on univariate natural exponential families due to C.N. Morris, which we apply here for the first time in the context of microscopic stochastic systems. In particular, we introduce a new model whose invariant measure is given by the generalized hyperbolic secant distribution. Additionally, under the same conditions, we show that, regardless of the specific model, the hydrodynamic equation is always the classical heat equation, with a diffusion coefficient that depends on the model. Our proof is based on deriving uniform bounds on second-moments of state variables, whose proof is achieved by relating a correlation function to a one-dimensional random walk whose jump rates are model-dependent and obtaining sharp bounds on its occupation times on specific domains.