Curves homogeneous under analytic transformations

We call a subset $K$ of $\mathbb C$ \emph{biholomorphically homogeneous} if for any two points $p,q\in K$ there exists a neighborhood $U$ of $p$ and a biholomorphism $\psi:U\to \psi(U)\subset \mathbb C$ such that $\psi(p)=q$ and $\psi(K\cap U)= K\cap \psi(U)$. We show that a biholomorphically homogeneous smooth curve $\gamma\subset \mathbb C$ is necessarily real-analytic. Furthermore we show that the same holds for the homogeneity with respect of a wider class of groups $G$ of real-analytic transformations of the plane. The result also extends to subsets $K\subset \mathbb R^2$ which are just locally closed.