Abstract: We consider a matrix periodic elliptic operator in a multidimensional space with distant perturbations. The order of unperturbed is even and arbitrary, while the perturbations are described by arbitrary abstract localized operators. The number of perturbations is abitrary but fixed. The main result is the explicit formula for the resolvent of the perturbed operator. This formula also allows to represent the resolvent as a convergent asymptotic series. In addition to the general result, we give a series of examples of both the unperturbed operator and perturbations.