On eigenvalues of the Landau Hamiltonian with a periodic electric potential

We consider the Landau Hamiltonian ${\widehat{H}}_B+V$ on $L^2({\mathbb{R}}^2)$ with a periodic electric potential $V$. For every $m\in \mathbb{N}$ we prove that there exist nonconstant periodic electric potentials $V\in C^{\mathrm{\infty }}({\mathbb{R}}^2;\mathbb{R})$ with zero mean values that analytically depend on a small parameter $\epsilon \in \mathbb{R}$ such that the Landau level $(2m+1)B$ is an eigenvalue of the Hamiltonian (of infinite multiplicity) where $B>0$ is a strength of a homogeneous magnetic field.