We consider the planar polynomial vector field $$(*) \;\;\;\begin{cases} \dot x= P(x,y) \newline \dot y =Q(x,y)\end{cases}$$
We replace the real variables $x,y$ with complex variables $x:=x_{1}+ix_{2},\;\;\; y:=y_{1}+iy_{2}$.
This gives us a real polynomial vector field in $\mathbb{R}^{4}$. This $4$ dimensional system is denoted by $(*_{C})$
Moreover, in the planar system $(*)$, as in the following post, we replace the real variables $x,y$ with square matrices $x,y \in M_{n}(\mathbb{R})$. This gives us a vector field on $M_{n}(\mathbb{R})\times M_{n}(\mathbb{R})$ which is denoted by $(*_{M})$.
Is there a planar polynomial vector field $(*)$ such that the corresponding higher dimensional system $(*_{C})$ or $(*_{M})$ has a compact invariant set $K$ which does not contain any singular point or periodic orbit?
In the other words:
Is the Poincare Bendixson theorem true for complexification or matrix form of a polynomial vector field on $\mathbb{R}^{2}?$
Note: I hope that the tag "index theory" I choose for this question, is not inappropriate. I choose it since I could not find a tag "Conley index". On the other hand this question may be some how related to methods in Conley index theory.
