Let $\{\mathbb{P}_{\theta}\}_{\theta}$ be an exponential family of probability measures on $\mathbb{R}^n$ with $$ \frac{d\mathbb{P}_{\theta}}{d\mathbb{P}} \propto \exp\left( \sum_{n=1}^N \eta_n(\theta) S_n(x) \right), $$ where $S_n:\mathbb{R}^n\rightarrow \mathbb{R}$ are its sufficient statistics.
Let $(X_t)$ be the Levy process with increments distributed according to this exponential family. Can we say anything about its Levy-Khintchine representation?