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.