We study the Schrödinger operator H = -Δ + 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 exp (i〈k,x〉) at the high energy region. Second, the isoenergetic curves in the space of momenta k corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous. © 2013 Hebrew University Magnes Press.