Invariants of some abelian $p$-groups in characteristic $p$

An explicit description of the ring of polynomial invariants of cyclic groups of order $p^s$ with fixed point set of codimension 2, or with covariants of codimension 2, over a field of characteristic $p$ is given. It transpires that these rings are complete intersections. A slight generalization for some abelian $p$-groups is also derived, which leads to a result about arbitrary groups with $p$-Sylow subgroup of this type.