  A new tool for the model theory of differentially closed fields and of
compact complex manifolds is here developed. In such settings, it is shown that
a type internal to the field of constants (resp. to the projective line) admits
a maximal image whose binding group is an abelian variety. The properties of
such "abelian reductions" are investigated in the Galois-theoretic framework
provided by stability theory.
  Several geometric consequences for the birational geometry of algebraic
vector fields of characteristic zero are then deduced. In particular, (1) it is
shown that if some cartesian power of an algebraic vector field admits a
nontrivial rational first integral then already the second power does, (2)
two-dimensional isotrivial algebraic vector fields are classified up to
birational equivalence, and (3) algebraic vector fields whose finite covers
admit no nontrivial factors are studied in arbitrary dimension.
  Analogues of these results in bimeromorphic geometry are also obtained.
