  We characterize Lie -Backlund vector fields in infinite dimensional jet
bundles $J^\infty(\mathbb{R}^n, \mathbb{R}^m)$ that can be exponentiated to
flows with each component depending on a finite set of variables. We show that
for $m=1$ each such field is an extension of one in a finite dimensional jet
space. For $m>1$ this is no longer the case and we give necessary and
sufficient conditions for exponentiation. Non-trivial examples are provided for
$(n,m)=(1,2)$.
