Abstract. We study the Rellich type theorem (RT) for the Maxwell operator Ĥ D = D̂Ĥ0 on Z3 with constant anisotropic medium, i.e. the permittivity and permeability of which are non-scalar constant diagonal matrices. We also study the unique continuation theorem for the perturbed Maxwell operator Ĥ Dp = D̂p Ĥ0 on Z3 where the permittivity and permeability are locally perturbed from a constant matrix on a compact set in Z3 . It then implies that, if Ĥ D − λ satisfies (RT), then all distributions û in Besov space B0∗ (Z3 ; C3 ) satisfying the equation (Ĥ Dp − λ)û = 0 outside a compact set vanish near infinity.