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Let $X$ be a real analytic vector field on $S^{2}$ which satisfies:

1. $X$ has a finite number of singularities on $S^{2}$

2. The equator is invariant under flow of $X$

3.$g_{*}X=X\; \text{or}=-X$ where $g$ is the antipodal map.

Can we say that $X$ is topological equivalent to the Poincare compactification of a polynomial vector field on $\mathbb{R}^{2}$?

This is motivated by the holomorphic situation:Every one dimensional singular holomorphic foliation of $\mathbb{C}P^{2}$ arise from a polynomial vector field(or polynomial one form) on $\mathbb{C}^{2}$. See "Dynamics of singular holomorphic foliations by curves" by Saeid Zakeri, Contem. Math $\;\;\;$ The first version at http://arxiv.org/pdf/math/9809099.pdf

Let $X$ be a real analytic vector field on $S^{2}$ which satisfies:

1. $X$ has a finite number of singularities on $S^{2}$

2. The equator is invariant under flow of $X$

3.$g_{*}X=\pm X$ where $g$ is the antipodal map $g(x,y,z)=(-x,-y,-z)$

Can we say that $X$ is topological equivalent to the Poincare compactification of a polynomial vector field on $\mathbb{R}^{2}$?

This is motivated by the holomorphic situation:Every one dimensional singular holomorphic foliation of $\mathbb{C}P^{2}$ arise from a polynomial vector field(or polynomial one form) on $\mathbb{C}^{2}$. See "Dynamics of singular holomorphic foliations by curves" by Saeid Zakeri, Contem. Math $\;\;\;$ The first version at http://arxiv.org/pdf/math/9809099.pdf

Let $X$ be a real analytic vector field on $S^{2}$ which satisfies:

1. $X$ has a finite number of singularities on $S^{2}$

2. The equator is invariant under flow of $X$

3.$g_{*}X=X\; \text{or}=-X$ where $g$ is the antipodal map.

Can we say that $X$ is topological equivalent to the Poincare compactification of a polynomial vector field on $\mathbb{R}^{2}$?

This is motivated by the holomorphic situation:Every one dimensional singular holomorphic foliation of $\mathbb{C}P^{2}$ arise from a polynomial vector field(or polynomial one form) on $\mathbb{C}^{2}$. See "Dynamics of singular holomorphic foliations by curves" by Saeid Zakeri, Conem. Math

Let $X$ be a real analytic vector field on $S^{2}$ which satisfies:

1. $X$ has a finite number of singularities on $S^{2}$

2. The equator is invariant under flow of $X$

3.$g_{*}X=X\; \text{or}=-X$ where $g$ is the antipodal map.

Can we say that $X$ is topological equivalent to the Poincare compactification of a polynomial vector field on $\mathbb{R}^{2}$?

This is motivated by the holomorphic situation:Every one dimensional singular holomorphic foliation of $\mathbb{C}P^{2}$ arise from a polynomial vector field(or polynomial one form) on $\mathbb{C}^{2}$. See "Dynamics of singular holomorphic foliations by curves" by Saeid Zakeri, Contem. Math $\;\;\;$ The first version at http://arxiv.org/pdf/math/9809099.pdf

Let $X$ be a real analytic vector field on $S^{2}$ which satisfies:

1. $X$ has a finite number of singularities on $S^{2}$

2. The equator is invariant under flow of $X$

3.$g_{*}X=X\; \text{or}=-X$ where $g$ is the antipodal map.

Can we say that $X$ is topological equivalent to the Poincare compactification of a polynomial vector field on $\mathbb{R}^{2}$?

This is motivated by the holomorphic situation:Every one dimensional singular holomorphic foliation of $\mathbb{C}P^{2}$ arise from a polynomial vector field(or polynomial one form) on $\mathbb{C}^{2}$

Let $X$ be a real analytic vector field on $S^{2}$ which satisfies:

1. $X$ has a finite number of singularities on $S^{2}$

2. The equator is invariant under flow of $X$

3.$g_{*}X=X\; \text{or}=-X$ where $g$ is the antipodal map.

Can we say that $X$ is topological equivalent to the Poincare compactification of a polynomial vector field on $\mathbb{R}^{2}$?

This is motivated by the holomorphic situation:Every one dimensional singular holomorphic foliation of $\mathbb{C}P^{2}$ arise from a polynomial vector field(or polynomial one form) on $\mathbb{C}^{2}$. See "Dynamics of singular holomorphic foliations by curves" by Saeid Zakeri, Conem. Math