Geodesibility of algebrizable three-dimensional vector fields

Recently, the geodesibility of planar vector fields, which are algebrizable (differentiable in the sense of Lorch for some associative and commutative unital algebra), has been established. In this paper, we consider algebrizable three-dimensional vector fields, for which we give rectifications and Riemannian metrics under which they are geodesible. Furthermore, for each of these vector fields $F$ we give two first integrals $h_1$ and $h_2$ such that the integral curves of $F$ are locally defined by the intersections of the level surfaces of $h_1$ and $h_2$.