I know that certain probability distributions may be derived from the requirement that entropy be maximal along with a constraint such as fixed variance. In the case of fixed variance, for example, one finds the normal distribution. In particular, the maximisation is over the set of all (!) continuous PDFs with that fixed variance.
Now my question is, is there a similarly general derivation of the Poisson distribution as a maximum entropy distribution? E.g. fixing that mean and variance are equal and maximising entropy? I have found a couple of articles but they always seem to prove maximality on a restricted set of discrete PDFs. Is it because there is no more general maximum entropy principle for the Poisson distribution? If so, is it because the discrete case is simply more complex than the continuous one?