  We prove a monodromy theorem for local vector fields belonging to a sheaf
satisfying the unique continuation property. In particular, in the case of
admissible regular sheaves of local fields defined on a simply connected
manifold, we obtain a global extension result for every local field of the
sheaf. This generalizes previous works of Nomizu for semi-Riemannian Killing
fields, of Ledger--Obata for conformal fields, and of Amores for fields
preserving a $G$-structure of finite type. The result applies to Finsler or
pseudo-Finsler Killing fields and, more generally, to affine fields of a spray.
Some applications are discussed.
