Proposition: Projected fixed points do not determine the exact verdict

For every $N\ge 1$, a smooth flow on ${\mathbb{R}}^{N+1}$ can have a projected fixed point in its first $N$ coordinates but no full fixed point. Two flows can also have the same projection and different dimensions of their ultraviolet-attracted families.

Consequently, a critical-family dimension claim requires a theorem about the exact flow and essential quotient, not only an eigenvalue count in a truncation.

Proof. Write $(x,y)\in {\mathbb{R}}^N\times \mathbb{R}$. The vector field

has no zero, while its projected flow $-x$ has fixed point zero.

Next compare

Both project to $-x$ and have fixed point $(0,0)$. Every trajectory of the first approaches it, whereas approach for the second requires $y=0$. Their attracted families have dimensions $N+1$ and $N$.