Topology on integers making it a topological group

Are there non-trivial topologies (neither discrete nor indiscrete) on the additive group of integers $\mathbb{Z}$, making it into a topological group. Could someone list them all, possibly with some details?

For any set $S$ of primes, the embedding

$$\mathbb{Z} \to \prod_{p \in S} \mathbb{Z}_p$$

of $\mathbb{Z}$ into the corresponding product of rings of $p$-adic integers gives a nontrivial topology on $\mathbb{Z}$ generated by arithmetic progressions of length $d$ where all prime divisors of $D$ lie in $S$. All of these topologies are distinct. When $S$ is the entire set of primes, we get the topology induced from the profinite completion, notably used in Fürstenburg's proof of the infinitude of the primes.

More generally one can take the initial topology with respect to any collection of quotients $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ where $n$ is chosen from a set $S$ of positive integers closed under taking divisors and $\text{lcm}$s. (It is generated by arithmetic progressions of length chosen from $S$.)