Integration of first-order ODEs by Jacobi fields

A new class of vector fields enabling the integration of first-order ordinary differential equations (ODEs) is introduced. These vector fields are not, in general, Lie point symmetries. The results are based on a relation between 2-dimensional Riemannian manifolds and the integrability of first-order ODEs, which was established in a previous work of the authors. An integration procedure is provided, together with several examples to illustrate it. A connection between integrating factors of first-order ODEs and Schr\"odinger-type equations is highlighted.