Is there a lie algebra structure $ [ \;. ] $ on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(\mathbb{S}^{2})$ which is not isomorphic to the standard structures but satisfies the following:
For every two vector fields $X,Y$ with $[X,Y]=0$, every limit cycle of $X$ must be invariant under $Y$.
The motivation is the following:
The usual Lie algebra structures satisfies the above dynamical property, but unfortunately a generic vector field has trivial centralizer in the standard Lie algebra structures. So we hope that for this (possible) new structure the centralizers are rich.(Are not trivial).
Note that a limit cycle for a vector field is an isolated closed orbit.