On the complex moment problem on radial rays

The complex moment problem in the case when the support of the measure lies on the algebraic curves $P_N=\{z\in\mathbb C:z^N-\bar z^N=0\}$, $N=1,2,3,\dots$, is studied. For $N=2,3$ the necessary and sufficient conditions of solvability are obtained and all solutions of the problem are described. It is shown how this problem for arbitrary $N$ is connected with the Hamburger moment problem with parameters.