We show that if the generalized variance of an infinitely divisible natural exponential family [math] in a [math] -dimensional linear space is of the form [math] , then there exists [math] in [math] such that [math] is a product of [math] univariate Poisson and ( [math] )-variate Gaussian families. In proving this fact, we use a suitable representation of the generalized variance as a Laplace transform and the result, due to Jörgens, Calabi and Pogorelov, that any strictly convex smooth function [math] defined on the whole of [math] such that [math] is a positive constant must be a quadratic form.