This is a constructivist question.
Please provide a bi-variate distribution or density/mass function of two absolutely continuous/discrete (but not mixed-type) random variables, which (may) have covariance zero and still be dependent.
But mind the constraints: The density/mass function should not have branches, i.e. each has to be a single mathematical expression for all the support of the distribution, where it is not zero. The same should hold for the marginal distributions. (In a parametric context, I guess the ideal will be to have a parameter governing covariance, and some other parameter(s) governing any other form of dependence, but "hard-wired" zero-covariance is ok for starters). And it would be preferable to avoid exotic supports.
Notes: It may be the case that this is trivially answered using the concept of Copulas. But we do want the marginals here. So any Copula-based answer I hope will be kind enough to explicitly provide also the marginals (or an example of marginals, if the answer provides a general mechanism of distributions-generation). Also, situations like e.g. a Bernoulli whose probability parameter is modeled as random and drawn from some other distribution, are not of interest here.
I am asking this and placing all these restrictions because to date, I find all examples of "zero covariance does not imply independence" absolutely true but artificial. I am thinking about situations where your "everyday" statistics practitioner (not necessarily a high-profile academic/professional) will be called upon to model -and most likely he will tend to try first some "smooth" representation. Does he have the theoretical modeling tools that will permit him to conclude "zero-covariance together with dependence"?
Naturally, I did try but have not yet found any such example -but this only attests to my knowledge gaps, not to whether such bi-variate distributions are hard to find. Meaning, any literature references are also welcome.