This paper focuses on the design of minimum-cost networks satisfying two technical constraints. First, the connected components should be unicyclic. Second, some given special nodes must belong to cycles. This problem is a generalization of two known problems: the perfect binary 2-matching problem and the problem of computing a minimum-weight basis of the bicircular matroid. It turns out that the problem is polynomially solvable. An exact extended linear formulation is provided. We also present a partial description of the convex hull of the incidence vectors of these Steiner networks. Polynomial-time separation algorithms are described. One of them is a generalization of the Padberg-Rao algorithm to separate blossom inequalities