$p$-elementary non-cyclic subgroups of the Cremona group of the plane

We classify, up to conjugacy, the subgroups of the Cremona group of the plane isomorphic to $(\mathbb{Z}/p\mathbb{Z})^r$, where $p$ is prime and $r \geq 2$, over an algebraically closed field $\mathbf{k}$ of characteristic not equal to $p$. In particular, we show that $r \leq 2$ if $p \geq 5$, $r \l