Notations : Suppose V is a closed contact compact manifold with contact form $\alpha$, of dimension 2n+1. Consider the symplectic sub-bundle $ \xi \subset TV $ given by $ \xi=$ ker($\alpha$). So $ \xi \rightarrow V $ is a vector bundle of rank $2n$ admitting the symplectic structure $d\alpha$. Also consider the Reeb vector field "$R$" determined by $\alpha$ and corresponding flow $\phi_t$. Assume that there is a finite number of simple periodic orbits of Reeb flow and all of them are non-degenerate.
Here is the question : In general there are plenty of almost complex structure on the symplectic bundle $\xi$ but is there any one which is invariant under Reeb flow, $\phi_t$ ?
I believe the answer will be no in general due to complicated behavior of $\phi_t$, but I wish that I am not right.