This is an an nouncement of the following results. We consider the Schrödinger operator H=−Δ+V(x) in dimension two, V(x) being a limitperiodic 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 ei(k,x)at the high energy region. Second, the isoenergetic curves in the space of momenta kk 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.