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).

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

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

Are  there  some researches   about this  dynamical system?Are there  closed orbits for  this  equation?The  latter is  equivalent  to ask: "Are there  zeroes  of  the  Riemann  Zeta function whose  Taylor  expansion (after  translation to the  origin  and real  rescalling ) is   in the  form $"iz+...."$. Every  zero  of  a  holomorphic  map  with this linear part    is  necessarily a center,  a  singularity surrounded  by  a  band  of  closed orbits.