Edit: According to the comment of Jaap Eldering, I realize that the previous version of my question was elementary, so I reformulate my question. I thank him for his comment.

Is there a real analytic vector field $X$, locally defined around $0\in \mathbb{R}^{2n}$ with the following properties:

1)The origin is an isolated singularity for $X$ and its linear part is the matrix $J=\begin{pmatrix}0&I\\-I&0 \end{pmatrix}$.

2)There is a Riemannian metric locally defined on a neighborhood $U$ of origin such that all trajectories of $X$ in $U\setminus \{0\}$ are unparametrized geodesic.

This question is some how a singular version of the concept "geodesible flow", a concept which is mainly related to Finding a 1-form adapted to a smooth flow, such that the flow has a unique isolated singularity whose linear part is in the form $\begin{pmatrix}0&I\\-I&0 \end{pmatrix}$.

The Motivation: There is no such metric for $n=1$.

Is there a real analytic vector field $X$, locally defined around $0\in \mathbb{R}^{2n}$, with the following properties:

1. The origin is an isolated singularity for $X$ and its linear part is the matrix $J=\begin{pmatrix}0&I\\-I&0 \end{pmatrix}$.

2. There is a Riemannian metric locally defined on a neighborhood $U$ of the origin, such that all trajectories of $X$ in $U\setminus \{0\}$ are unparametrized geodesic.

This question is somehow a singular version of the concept "geodesible flow", a concept which is mainly related to Finding a 1-form adapted to a smooth flow, such that the flow has a unique isolated singularity whose linear part is in the form $\begin{pmatrix}0&I\\-I&0 \end{pmatrix}$.

Motivation: There is no such metric for $n=1$.

Edit: According to the comment of Jaap Eldering, I realized that the previous version of my question was elementary, so I've reformulated my question. I thank him for his comment.

Edit: According to the comment of Jaap Eldering, I realize the previous version of my question was elementary, so I reformulate my question.

Is there a real analytic vector field $X$, locally defined around $0\in \mathbb{R}^{2n}$ with the following properties:

1)The origin is an isolated singularity for $X$ and its linear part is the matrix $J$, the standard complex structure of $\mathbb{R}^{2n}$

2)There is a Riemannian metric locally defined on a neighborhood $U$ of origin such that all trajectories of $X$ in $U\setminus \{0\}$ are unparametrized geodesic.

This question is some how a singular version of the concept "geodesible flow", a concept which is mainly related to Finding a 1-form adapted to a smooth flow, such that the flow has a unique isolated singularity whose linear part is in the form $\begin{pmatrix}0&I\\-I&0 \end{pmatrix}$.

The Motivation: There is no such metric for $n=1$.

Edit: According to the comment of Jaap Eldering, I realize that the previous version of my question was elementary, so I reformulate my question. I thank him for his comment.

Is there a real analytic vector field $X$, locally defined around $0\in \mathbb{R}^{2n}$ with the following properties:

1)The origin is an isolated singularity for $X$ and its linear part is the matrix $J=\begin{pmatrix}0&I\\-I&0 \end{pmatrix}$.

2)There is a Riemannian metric locally defined on a neighborhood $U$ of origin such that all trajectories of $X$ in $U\setminus \{0\}$ are unparametrized geodesic.

This question is some how a singular version of the concept "geodesible flow", a concept which is mainly related to Finding a 1-form adapted to a smooth flow, such that the flow has a unique isolated singularity whose linear part is in the form $\begin{pmatrix}0&I\\-I&0 \end{pmatrix}$.

The Motivation: There is no such metric for $n=1$.

Edit: According to the comment of Jaap Eldering, I realize the previous version of my question was elementary, so I reformulate my question.

Is there a real analytic vector field $X$, locally defined around $0\in \mathbb{R}^{2n}$ with the following properties:

1)The origin is an isolated singularity for $X$ and its linear part is the matrix $J$, the standard complex structure of $\mathbb{R}^{2n}$

2)There is a Riemannian metric locally defined on a neighborhood $U$ of origin such that all trajectories of $X$ in $U\setminus \{0\}$ are unparametrized geodesic.

This question is some how a singular version of the concept "geodesible flow" which is related to the following post: Finding a 1-form adapted to a smooth flow  

Edit: According to the comment of Jaap Eldering, I realize the previous version of my question was elementary, so I reformulate my question.

Is there a real analytic vector field $X$, locally defined around $0\in \mathbb{R}^{2n}$ with the following properties:

1)The origin is an isolated singularity for $X$ and its linear part is the matrix $J$, the standard complex structure of $\mathbb{R}^{2n}$

2)There is a Riemannian metric locally defined on a neighborhood $U$ of origin such that all trajectories of $X$ in $U\setminus \{0\}$ are unparametrized geodesic.

This question is some how a singular version of the concept "geodesible flow", a concept which is mainly related to Finding a 1-form adapted to a smooth flow, such that the flow has a unique isolated singularity whose linear part is in the form $\begin{pmatrix}0&I\\-I&0 \end{pmatrix}$.

The Motivation: There is no such metric for $n=1$.