Let $A = \mathbb{Z}[x, x^{-1}, y, y^{-1}]$. How can one find the ring of invariants $A^G$ for finite subgroups $G \subset \text{GL}(2,\mathbb{Z})$?
The action of a matrix $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$ is described by $x \mapsto x^a y^c$ and $y \mapsto x^b y^d$.
For example, let $G \cong D_3$, where the isomorphism is given by \begin{align*} s & \mapsto \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \\ r & \mapsto \begin{pmatrix} 0 & 1 \\ -1 & -1 \end{pmatrix}, \end{align*} act on $A$.
Is there some nice software to solve such problems?
EDIT: Invariant rings, produced by reflection groups, can be polynomial rings. Is there an algorithm for finding the generators of such polynomial rings (for two and more dimensions)?