Let 𝒟 be a simply connected
plane domain and let r be a positive integer or ∞. By a Cr direction field Φ on
𝒟 we mean Cr mapping Φ : 𝒟→ G2.1, the projective line consisting of
lines through the origin, 0, in the plane. A Cr first integral of Φ is a Cr
function f : 𝒟→ R such that each level set of f has no interior and is a
union of members of the family, ℱ, of maximal integral curves of Φ. We
show, in general, that first integrals do not exist and then give a necessary
and sufficient condition for a Cr first integral to exist. When Φ has a first
integral we also show that there exists a local diffeomorphism μ : 𝒟→ R2
such that Φ is mapped by μ into the (constant) vertical direction field on
μ(𝒟).
