Consider a market with a monopolist firm that has zero marginal cost and faces demand $D(p;\mathbf{a})$, where $\mathbf{a}$ is a vector of parameters and $p$ is the price. The monopolist maximizes profit by solving $$\max_p D(p;\mathbf{a})p,$$ so that the optimal price, $p^*$, satisfies $$D_1(p^*;\mathbf{a})p^*+D(p^*;\mathbf{a})=0.$$
This optimal price, $p^*$, results in consumer surplus $$\text{CS}=\int_{p^*}^\infty\!D(p;\mathbf{a})\,dp,$$ and producer surplus $$\text{PS}=D(p^*,\mathbf{a})p^*.$$
My question is: is there a family of demand functions, $D(p;\mathbf{a})$, such that $\text{CS}=\text{PS}$ always holds at $p^*$, and if so what does the functional form look like?