The Poincaré-Bendixson theorem completely characterizes the $ \omega$ -limit sets of planar systems.

I would like to know whether extensions exist to 3D systems which tend to 2D systems in the following sense: Suppose that as the independent variable increases, the motion approaches a plane, as in the following system: $ $ \begin{align} \dot{x}&=f(x,y,z), \ \dot{y} &=g(x,y,z), \ \dot{z} &=-z. \end{align} $ $ The (bounded) $ \omega$ -limit set of any point are necessarily on the plane $ \{z=0 \}$ , and they are invariant sets of the flow $ $ \begin{align} \dot{x} &= f(x,y,0), \ \dot{y} &= g(x,y,0). \end{align} $ $ Must these $ \omega$ -limit sets be, as in the conclusion of Poincaré-Bendixson, be:

- a fixed point
- a periodic orbit or
- a connected set composed of a finite number of fixed points together with homoclinic and heteroclinic orbits connecting these?

Thank you!