The paper is devoted to divergence-curl results involving a divergence free measurevalued field σ = ν b, where ν is a signed Radon measure on R N and b is a non-vanishing regular vector field in R N , and a gradient measure-valued field η = ∇u on R N , N ≥ 2. On the one hand, in a non-periodic framework we prove that for any open set Ω of R 2 , the orthogonality condition b • ∇u = 0 in Ω implies the equality div (u σ) = 0 in Ω. The key-ingredient of the proof is based on the existence of a representative in L ∞ loc (Ω) of the bounded variation function u in Ω. This result allows us to extend in the setting of ODE's flows, the famous Franks-Misiurewicz theorem which claims that the Herman rotation set of any continuous two-dimensional flow on the torus T 2 is a closed line segment of a line of R 2 passing through 0 R 2. Moreover, this non-periodic divergence-curl result can be applied to a finite almost periodic bounded variation function u and to a finite almost periodic measure-valued field σ = ν b. On the other hand, in the periodic case with dimension N > 2, assuming that ν is absolutely continuous with respect to Lebesgue's measure on the torus T N , we prove that if the product b • ∇u is the zero measure on T N , so is the product of the T N-means ν b • ∇u.