**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.