  We study when a given edge of a factor-critical graph is contained in a
matching avoiding exactly one, pregiven vertex of the graph. We then apply the
results to always partition the vertex-set of a $3$-regular, $3$-uniform
hypergraph into at most one triangle (hyperedge of size $3$) and edges (subsets
of size $2$ of hyperedges), corresponding to the intuition, and providing new
insight to triangle and edge packings of Cornu\'ejols' and Pulleyblank's. The
existence of such a packing can be considered to be a hypergraph variant of
Petersen's theorem on perfect matchings, and leads to a simple proof for a
sharpening of Lu's theorem on antifactors of graphs.
