The canonical way to extend Peano arithmetic to the integers is not by changing the language or axioms, but instead by treating integers as equivalence classes of pairs of natural numbers.

We think of an integer $x$ as any pair $(a,b)$ of natural numbers with $a-b=x$. More formally, we translate arithmetic statements about integers by first replacing the integers with pairs of natural numbers, and then transforming the results by
\begin{align}
0_Z& \to (0,0)\\
S(a,b)& \to (Sa,b)\\
(a,b)+(c,d)& \to (a+c,b+d)\\
(a,b)(c,d)& \to (ac+bd,ad+bc)\\
(a,b)=(c,d)& \to a+d=b+c
\end{align}

This process turns every arithmetic statement about integers into an arithmetic statement about natural numbers instead. Eg $$(\exists x,y,z\in\mathbb{Z})x^3+y^3+z^3=30$$
can be translated idiomatically (i.e. after performing the mechanical translation and then some of the usual simplifications) as
\begin{align}(\exists a,b,c,d,e,f)& a^3+3ab^2+c^3+3cd^2+e^3+3ef^2=\\
&b^3+3ba^2+d^3+3dc^2+f^3+3fe^2+30\end{align}