I got some trouble understanding La Salles invariance principle, more specifically this exercise where I want to investigate stability of the origin given system
$$\begin{cases} x'=-y-x^3\\ y'=x^5 \end{cases}$$
I've chosen test function $V(x,y)=x^6+\alpha y^2$ which gives that $V_f(z)=-6x^8<0$ with the right choice of $\alpha$
Now to the part I don't understand. The solution from the course literature says that we check invariant sets of the system of the set $V_f^{-1}(0)$ (Where $V_f(x,y)=0$). They come to the conclusion that the origin is the only invariant set on $V_f^{-1}(0)$, which I don't understand.
A invariant set is a set for which we start in, and stay in as time goes along. This, to me, means that the whole set $V_f^{-1}(0)$ should be an invariant set itself, since it is 0?
Would anyone mind explaining to me why the only invariant set in this set is the origin?
Thanks!