Is there any example of curved exponential family with $T$ being a complete statistic? Here $T$ is the sufficient statistic.
Besides the bivariate Normal example, Keener gives an exercise in relation with the question:
2. When $X\sim \mathcal B(p)$ and $Y\sim\mathcal B(h(p))$, the joint distribution of $(X,Y)$ is curved except for a specific function $h_0(p)$. Give two functions $h(p)$ for which $(X,Y)$ is minimal sufficient but not complete and for which $(X,Y)$ is complete.