This paper gives a characterization of valuations that follow the singular infinitely near points of plane vector fields, using the notion of L'H\^opital valuation, which generalizes a well known classical condition. With that tool, we give a valuative description of vector fields with infinite solutions, singularities with rational quotient of eigenvalues in its linear part, and polynomial vector fields with transcendental solutions, among other results.

Abelian closed subgroups of the Galois group of the pythagorean closure of a formally real field are described by means of the inertia group of suitable valuation rings.