The normal, Poisson, gamma, binomial, and negative binomial distributions are univariate natural exponential families with quadratic variance functions (the variance is at most a quadratic function of the mean). Only one other such family exists. Much theory is unified for these six natural exponential families by appeal to their quadratic variance property, including infinite divisibility, cumulants, orthogonal polynomials, large deviations, and limits in distribution.
Carl N. Morris. "Natural Exponential Families with Quadratic Variance Functions." Ann. Statist. 10 (1) 65 - 80, March, 1982. https://doi.org/10.1214/aos/1176345690