Atoms of root-closed submonoids of $\mathbb{Z}^2$

We describe how one can explicitly obtain all atoms of an arbitrary root-closed monoid, whose quotient group is isomorphic to $\mathbb{Z}^2$. For this purpose, we solve this task for three special types of such monoids in Theorems 5 and 6, and then transfer these results to the general case. It turns out that all atoms can be obtained from the (regular) continued fraction expansion of the slopes of the bounding rays of the cone, which is spanned by the monoid.