  Lov\'asz and Cherkasskiy discovered in the 1970s independently that if $ G $
is a finite graph with a given set $ T $ of terminal vertices such that $ G $
is inner Eulerien, then the maximal number of edge-disjoint paths connecting
distinct vertices in $ T $ is $ \sum_{t\in T}\lambda(t, T-t) $ where $\lambda $
is the local edge-connectivity function. The maximal size of a system of
edge-disjoint $ T $-paths in the Lov\'asz-Cherkasskiy theorem can be
characterized by the existence of certain cuts applying Menger's theorem. The
infinite generalization of Menger's theorem by Aharoni and Berger (earlier
known as the Erd\H{o}s-Menger conjecture) together with the characterization of
infinite Eulerian graphs due to Nash-Williams make possible to generalize the
theorem for infinite graphs in a structural way. The aim of the paper is to
prove this generalization when $ T $ is countable.
