Let $\Gamma \subset \mathbb{C}$ be a smooth curve such that monomials are orthogonal on it, i.e. with $n,m \in \mathbb{N} \cup\{0\}$
$$\int_{\Gamma} z^n \overline{z^m} |dz| = 0, \qquad \qquad \forall n \neq m$$
An example of such curve is a circle around origin. Are there any other examples?