  We consider the Landau Hamiltonian $H_0$, self-adjoint in $L^2({\mathbb
R^2})$, whose spectrum consists of an arithmetic progression of infinitely
degenerate positive eigenvalues $\Lambda_q$, $q \in {\mathbb Z}_+$. We perturb
$H_0$ by a non-local potential written as a bounded pseudo-differential
operator ${\rm Op}^{\rm w}({\mathcal V})$ with real-valued Weyl symbol
${\mathcal V}$, such that ${\rm Op}^{\rm w}({\mathcal V}) H_0^{-1}$ is compact.
We study the spectral properties of the perturbed operator $H_{{\mathcal V}} =
H_0 + {\rm Op}^{\rm w}({\mathcal V})$. First, we construct symbols ${\mathcal
V}$, possessing a suitable symmetry, such that the operator $H_{\mathcal V}$
admits an explicit eigenbasis in $L^2({\mathbb R^2})$, and calculate the
corresponding eigenvalues. Moreover, for ${\mathcal V}$ which are not supposed
to have this symmetry, we study the asymptotic distribution of the eigenvalues
of $H_{\mathcal V}$ adjoining any given $\Lambda_q$. We find that the effective
Hamiltonian in this context is the Toeplitz operator ${\mathcal T}_q({\mathcal
V}) = p_q {\rm Op}^{\rm w}({\mathcal V}) p_q$, where $p_q$ is the orthogonal
projection onto ${\rm Ker}(H_0 - \Lambda_q I)$, and investigate its spectral
asymptotics.
