  An Euler tour in a hypergraph is a closed walk that traverses each edge of
the hypergraph exactly once, while an Euler family, first defined by Bahmanian
and Sajna, is a family of closed walks that jointly traverse each edge exactly
once and cannot be concatenated. In this paper, we study the notions of a
spanning Euler tour and a spanning Euler family, that is, an Euler tour
(family) that also traverses each vertex of the hypergraph at least once. We
examine necessary and sufficient conditions for a hypergraph to admit a
spanning Euler family, most notably, when the hypergraph possesses a vertex cut
consisting of vertices of degree two. Moreover, we characterise hypergraphs
with a vertex cut of cardinality at most two that admit a spanning Euler tour
(family). This result enables us to reduce the problem of existence of a
spanning Euler tour (which is NP-complete), as well as the problem of a
spanning Euler family, to smaller hypergraphs.
