  We study perfect multiple coverings in translation invariant graphs with
vertex set $\mathbb{Z}^2$ using an algebraic approach. In this approach we
consider any such covering as a two-dimensional binary configuration which we
then express as a two-variate formal power series. Using known results, we
conclude that any perfect multiple covering has a non-trivial periodizer, that
is, there exists a non-zero polynomial whose formal product with the power
series presenting the covering is a two-periodic configuration. If a
non-trivial periodizer has line polynomial factors in at most one direction,
then the configuration is known to be periodic. Using this result, we find many
setups where perfect multiple coverings of infinite grids are necessarily
periodic. We also consider some algorithmic questions on finding perfect
multiple coverings.
