This paper deals with the solvability of planar vector fields L = A∂x+B∂y, with A and B complex-valued function in a domain Ω ⊂ ℝ2. We assume that L has a first integral Z that is a homeomorphism in Ω. To such a vector field, we associate a Cauchy–Pompeiu type operator and investigate the Hölder solvability of Lu = f and of a related Riemann–Hilbert problem when f is in Lp with p>2+σ, where σ is a positive number associated to L.
