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This is not an answer, but is a comment. (I can not give comment since I am under 50 reputation).

Linear vector fields are always complete vector field so they does not satisfy your condition.

But for higher order polynomial vector field, I guess that the solutions which are not a complete orbits, are not in $L ^2$. My motivation is that according to an interesting Paper of Chicone and Sotomayor, the solutions escape at infinity very fast(exponentially) since there is a hyperbolic singularity at equator.

On the other hand your question is very interesting for me since it implicitly suggests to consider some different function spaces to be acted by $D_f$, the derivational operator associated to the vector field $f$.

The motivations for study of this derivational operator is explained in the following posts:

Does this function belong to $L^2(\mathbb{D})$?

Codimension of the range of certain linear operators

This is not an answer, but is a comment. (I can not give comment since I am under 50 reputation).

Linear vector fields are always complete vector field so they do not satisfy your condition.

But for higher order polynomial vector field, I guess that the solutions which are not a complete orbits, are not in $L ^2$. My motivation is that according to an interesting Paper of Chicone and Sotomayor, the solutions escape at infinity very fast(exponentially) since there is a hyperbolic singularity at equator.

On the other hand your question is very interesting for me since it implicitly suggests to consider some different function spaces to be acted by $D_f$, the derivational operator associated to the vector field $f$.

The motivations for study of this derivational operator is explained in the following posts:

Does this function belong to $L^2(\mathbb{D})$?

Codimension of the range of certain linear operators

This is not an answer, but is a comment. (I can not give comment since I am under 50 reputation).

Linear vector fields are always complete vector field so they does not satisfy your condition.

But for higher order polynomial vector field, I guess that the solutions which are not a complete orbits, are not in $\ell^2$. My motivation is that according to an interesting Paper of Chicone and Sotomayor, the solutions escape at infinity very fast(exponentially) since there is a hyperbolic singularity at equator.

On the other hand your question is very interesting for me since it implicitly suggests to consider some different function spaces to be acted by $D_f$, the derivational operator associated to the vector field $f$.

The motivations for study of this derivational operator is explained in the following posts:

Does this function belong to $L^2(\mathbb{D})$?

Codimension of the range of certain linear operators

This is not an answer, but is a comment. (I can not give comment since I am under 50 reputation).

Linear vector fields are always complete vector field so they does not satisfy your condition.

But for higher order polynomial vector field, I guess that the solutions which are not a complete orbits, are not in $L ^2$. My motivation is that according to an interesting Paper of Chicone and Sotomayor, the solutions escape at infinity very fast(exponentially) since there is a hyperbolic singularity at equator.

On the other hand your question is very interesting for me since it implicitly suggests to consider some different function spaces to be acted by $D_f$, the derivational operator associated to the vector field $f$.

The motivations for study of this derivational operator is explained in the following posts:

Does this function belong to $L^2(\mathbb{D})$?

Codimension of the range of certain linear operators

This is not an answer, but is a comment. (I can not give comment since I am under 50 reputation).

Linear vector field are always complete vector field.

But for higher order polynomial vector field, I guess that the solutions which are not a complete orbits, are not in $\ell^2$. My motivation is that according to a Paper of Chicone and Sotomayor, the solutions escape at infinity very fast(exponentially) since there is a hyperbolic singularity at equator. But your question is very interesting for me since it implicitelty suggest to consider some different function space for consideration of $D_f$ the derivational operator associated to the vector field.

This is not an answer, but is a comment. (I can not give comment since I am under 50 reputation).

Linear vector fields are always complete vector field so they does not satisfy your condition.

But for higher order polynomial vector field, I guess that the solutions which are not a complete orbits, are not in $\ell^2$. My motivation is that according to an interesting Paper of Chicone and Sotomayor, the solutions escape at infinity very fast(exponentially) since there is a hyperbolic singularity at equator.

On the other hand your question is very interesting for me since it implicitly suggests to consider some different function spaces to be acted by $D_f$, the derivational operator associated to the vector field $f$.

The motivations for study of this derivational operator is explained in the following posts:

Does this function belong to $L^2(\mathbb{D})$?

Codimension of the range of certain linear operators