Yulia Karpeshina, Young-Ran Lee Spectral properties of a limit-periodic Schr\"{o}dinger operator in dimension two (549K, LaTeX 2e with 5 EPS Figures) ABSTRACT. We study Schr\"{o}dinger operator $H=-\Delta+V(x)$ in dimension two, $V(x)$ being a limit-periodic potential. We prove that the spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i\langle ec k, ec x angle }$ at the high energy region. Second, the isoenergetic curves in the space of momenta $ ec k$ corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous.