Let us fix some smooth vector field $F(x)$ in some open submanifold $M\subseteq \mathbb R^n$, $n\ge3$. A vector field $G(x)$ is said to be compatible with $F(x)$ if the Lie bracket $[F,G]$ is a linear combination of $F(x)$ and $G(x)$, i.e. $$ \forall x\in M \quad [F(x),G(x)]= \alpha(x) F(x)+\beta(x) G(x), $$ where $\alpha(x)$ and $\beta(x)$ are some smooth scalar functions.
Denote by $C_F$ the set of all smooth vector fields compatible with a given vector field $F(x)$. What is the structure of the set $C_F$? Does it always contain any vector fields other than fields of the form $\gamma(x) F(x)$, where $\gamma(x)$ is a smooth scalar function? And if so, how can one get such vector fields?
The vector fields $F(x)$ and $G(x)$ determine an integrable distribution. So you would look for $2$-dimensional foliations of $M$ whose leaves contain the trajectories of $F$. That can be done in many ways...
Note: if $G(x)$ is such a vector field, and $\phi$ is a smooth function then $\phi \cdot G$ is again.
I am not sure that it is a vector space ( closed under addition). Have you checked that?