Is there a complete classification of quadratic polynomial vector fields on $\mathbb{C}^2$ whose corresponding singular foliation of $\mathbb{C}P^2$ satisfies the following:
The regular leaves of the foliation are totally geodesics 2 dimensional real submanifolds of the projective space endowed with the Fubini-Study metric?
Is there a complex quadratic vector field for which the corresponding singular foliation of projective space is not geodesible in the following sense:
A singular foliation of projective space is geodesible if there is a Riemannian metric defined on the whole space minus singularities such that the leaves of the foliation are totally geodesic. One can think to the later question without projectivization (working in $\mathbb{C}^2$).
The motivation for the later question is that a real quadratic vector field is always geodesible. Please see the following post:
