Let $\zeta$ be the classical Riemann zeta function.

We define a differential equation on $\mathbb{R}^{2} \setminus \{1\}$ by $\dot Z= \zeta(Z)$. From a foliation point of view this vector field can be counted as a smooth vector field on whole $\mathbb{R}^{2}$ with the following equivalent formulation(They have the same trajectories).

$$\dot Z= \parallel z-1\parallel^2 \zeta(Z)$$

Then the field has a saddle point at $1$.

Are there some researches about this dynamical system?Are there closed orbits for this equation?