  We consider congestion games on networks with nonatomic users and
user-specific costs. We are interested in the uniqueness property defined by
Milchtaich [Milchtaich, I. 2005. Topological conditions for uniqueness of
equilibrium in networks. Math. Oper. Res. 30 225-244] as the uniqueness of
equilibrium flows for all assignments of strictly increasing cost functions. He
settled the case with two-terminal networks. As a corollary of his result, it
is possible to prove that some other networks have the uniqueness property as
well by adding common fictitious origin and destination.
  In the present work, we find a necessary condition for networks with several
origin-destination pairs to have the uniqueness property in terms of excluded
minors or subgraphs. As a key result, we characterize completely bidirectional
rings for which the uniqueness property holds: it holds precisely for nine
networks and those obtained from them by elementary operations. For other
bidirectional rings, we exhibit affine cost functions yielding to two distinct
equilibrium flows. Related results are also proven. For instance, we
characterize networks having the uniqueness property for any choice of
origin-destination pairs.
