I tried Mobius transformations $ \frac{az+b}{cz+d}$ for several $ z∈ \mathbb{Z}+i\mathbb{Z}$ and it seems that even if $ M=\begin{pmatrix}a&b\c&d\end{pmatrix} \in SL_2(\mathbb{Z})$ , not any ‘integer-complex’ ($ z∈ \mathbb{Z}+i\mathbb{Z}$ ) transforms to a ‘integer-complex’ and vice-versa. If I am not mistaken after evaluation of $ \frac{az+b}{cz+d}$ even lines can be transformed to circles and and vice-versa.

My question is, for what subset of $ SL_2(\mathbb{Z})$ (answer should include cases when $ c=0$ and $ d=\pm1$ because this cases meet the requirements I think), a $ z∈ \mathbb{Z}+i\mathbb{Z}$ always is transformed to a $ w∈ \mathbb{Z}+i\mathbb{Z}$ by transformations $ \frac{az+b}{cz+d}$ ? In other words, a set that transforms lattices to lattices?