Abstract. For each algebrizable three-dimensional vector field F, in this paper we give local rectifications Hα of F, which let us to show that F is geodesible with respect to the Riemannian metric g. Furthermore, an orthonormal frame {E1,E2,E3} for g, where Ei = eiF and {e1,e2,e3} is the canonical basis of R3. This orthonormal frame extends to a family of vector fields F, defined by the product of F by the unitary vectors of R3, whose integral curves corresponds with unitary geodesics of g. We found three conservative vector fields Gi, i =1,2,3, such that 〈Ei,Gj〉 = δij with respect to the usual inner product of R3, where δij is the Kronecker Delta. Thus, we obtain two first integrals Φ1 and Φ2 of F, with corresponding level sets having transversal intersections that contain the integral curves of F. g give rise to a metric local metric dα.