# Singular foliations of $\mathbb{C}P^2$ that are compatible to Fubini-Study metric

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:

Finding a 1-form adapted to a smooth flow

• @Pedro Thank you for your very helpful edit. – Ali Taghavi Feb 1 '18 at 14:30