I have a matrix $A \in \mathbb Z^{n \times m}$, where $m > n$, and a vector $b \in \mathbb Z^n$. Under what conditions does

$$Ax = b$$

have an integer solution? Is there a way to bound the norm of the solution vector $x$ in terms of the norms of $A$ and $b$?

Essentially, I want something like Siegel's lemma, but for the non-homogeneous case.

I am not an expert on this and will appreciate any help. Thanks!