# Center manifold of sets of equilibria

My question is regarding Center Manifolds containing a continuous set of equilibrium points. The theory I have studied talks about the existence of a center manifold for equilibrium points, but what happens if we do not have an isolated equilibrium point but a continuous set of equilibrium points? Let me give a toy example:

$\dot x=-y^2\\ \dot y=-y^2x\\ \dot z=-z+y^2$

The $x$-axis is a continuous set of equilibrium points, the linear part is (all along this line) $A=\begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & -1 \end{bmatrix}$

So my interpretation is that there exists a 2-dimensional center manifold which contains the $x-$axis. Is this correct? Do you know a source for a proof?

Next if I would like to compute this center manifold? Does power series still apply? This is, can I propose an expression of the form:

$z=\displaystyle\sum_{i,j}a_{ij}x^iy^j$?

I guess it can´t be that "easy" since we would like to capture that the center manifold, at least in this case, is tangent to the 2-dimensional center space all along the $x-$axis. Anyway I am stuck here...

Thanks for any help in understanding this. Literature for reference is also appreciated.

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