  Tautological systems developed in [6],[7] are Picard-Fuchs type systems to
study period integrals of complete intersections in Fano varieties. We
generalize tautological systems to local complete intersections, which are zero
loci of global sections of vector bundles over Fano varieties. In particular,
we obtain similar criterion as [6, 7] about holonomicity and regularity of the
system. We also prove solution rank formulas and geometric realizations of
solutions following the work of hypersurfaces [5, 4].
