  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 $e^{i\langle \vec k,\vec x\rangle }$ at the high energy
region. Second, the isoenergetic curves in the space of momenta $\vec k$
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.
