Centralizers of elements of infinite order in plane Cremona groups

Let $\mathbf{K}$ be an algebraically closed field. The Cremona group $\operatorname{Cr}_2(\mathbf{K})$ is the of birational transformations projective plane $\mathbb{P}^2_{\mathbf{K}}$. We carry out overall study centralizers elements infinite order in which leads to a classification embeddings $\mathbf{Z}^2$ into $\operatorname{Cr}_2(\mathbf{K})$, as well maximal non-torsion abelian subgroups when $\operatorname{char}(\mathbf{K})=0$.