Neither stable or unstable equilibrium

Question

I found a non-linear dynamical system which has a line of equilibrium points at $y=0$; when linearizing and evaluating at those points I find that Jacobian matrix is J=$\begin{bmatrix}0 &1\\0&0\end{bmatrix}$ on the line; what can I say about the kind of equilibrium?

Answer 1

This indicates that there are no stable and no unstable dynamics around the critical point. You may want to look at the center manifold.