On the Novikov problem for dihedral symmetry potentials

We consider Novikov’s problem of describing level lines of quasiperiodic functions on a plane for two-dimensional potentials of dihedral symmetry. It is shown that quasiperiodic potentials of this type can have open level lines only at a single energy level $\epsilon ={\epsilon }_0$, which brings them close to random potentials on a plane.