What do you want? A linear system? It is boooring (and cannot be any other way)
$$
\begin{aligned}
&x' = x,\\
&y' =y
\end{aligned}
$$
Nonlinear system? Try this
$$
\begin{aligned}
&x' = x-.1y(x^2+y^2),\\
&y' = y+.1x(x^2+y^2)
\end{aligned}
$$
The variation appear as this eigenvalue is $0$. But then linear system is simply trivial
$$
\begin{aligned}
&x' = 0,\\
&y' =0
\end{aligned}
$$
but non-linear could be entertaining
$$
\begin{aligned}
&x' = (x-y)(x^2+y^2),\\
&y' =(x+y)(x^2+y^2)
\end{aligned}
$$
or
$$
\begin{aligned}
&x' = xy,\\
&y' =(x+y)(x^2+y^2)
\end{aligned}
$$
however nothing can be derived from linearization.
