## When an ‘integer-complex’ always is transformed to an ‘integer-complex’?

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?