The canonical way to extend Peano arithmetic to the integers is by treating integers as equivalence classes of pairs of integers,
$$(a,b)~(c,d)\leftrightarrow a+d=b+c$$
with the translations
\begin{align}
0_Z&=(0,0)\\
S(a,b)&=(Sa,b)\\
(a,b)+(c,d)&=(a+c,b+d)\\
(a,b)(c,d)&=(ac+bd,ad+bc)\end{align}

Then every arithmetic statement about integers becomes 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 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}