  Zero-dimensional $O(n)$-symmetric sigma models are studied by using
Picard--Lefschetz integration method in the presence of small symmetry-breaking
perturbations. Due to approximate symmetry, downward flows turn out to show
significant structures: They slowly travel along the set of pseudo classical
points, and branch into other directions so as to span middle-dimensional
integration cycles. We propose an efficient way to find such slow motions for
computing Lefschetz thimbles. In the limit of symmetry restoration, we figure
out that only special combinations of Lefschetz thimbles can survive as
convergent integration cycles: Other integrations become divergent due to
non-compactness of the complexified group of symmetry. We also compute downward
flows of $O(2)$-symmetric fermionic systems, and confirm that all of these
properties are true also with fermions.
