  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.
