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 property quoted below?

> The regular leaves of the foliation are totally geodesic 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*?

> *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 of 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:

https://mathoverflow.net/questions/273635/finding-a-1-form-adapted-to-a-smooth-flow/273648#273648