Difference between bivariate exponential family and bivariate elliptical family

Question

I'm studying the elliptical family and I'd like to know the principal differences between the bivariate exponential and elliptical family.

Answer 1

Following a result in Kelker (1970, Sankhya A), the Normal distribution is the only intersection between elliptical and exponential families. I recommend reading this fundamental paper about elliptical families, which are rarely covered by textbooks, as opposed to exponential families.