Here, I use Peano-like axioms to describe the set of integers $Z$. They are based on two successor functions, each starting with a common point of $0$, and a principle of induction for the integers.
Let $Z$, $Pos$, $Neg$, $s$, $s'$ and $0$ be such that:
$Z=Pos\cup Neg$ (edit)
$\forall x (x\in Pos \wedge x\in Neg \leftrightarrow x=0)$
$s$ is injective
$s'$ is injective
$\forall x\in Pos (s(x)\neq 0)$
$\forall x\in Neg (s'(x)\neq 0)$
$\forall m ((0\in m\wedge \forall x\in Pos (x\in m\rightarrow s(x)\in m) \wedge \forall x\in Neg (x\in m\rightarrow s'(x)\in m))\rightarrow \forall x\in Z (x\in m)) $
Note that, contrary to the usual convention, I have had to include $0$ in both sets $Pos$ and $Neg$.
Lemma: $0\in Z, Pos, Neg$
See my follow-up below