  We extend the classical Stallings theory (describing subgroups of free groups
as automata) to direct products of free and abelian groups: after introducing
enriched automata (i.e., automata with extra abelian labels), we obtain an
explicit bijection between subgroups and a certain type of such enriched
automata, which - as it happens in the free group - is computable in the
finitely generated case.
  This approach provides a neat geometric description of (even non finitely
generated) intersections of finitely generated subgroups within this non-Howson
family. In particular, we give a geometric solution to the subgroup
intersection problem and the finite index problem, providing recursive bases
and transversals respectively.
