The Lovász-Cherkassky theorem for locally finite graphs with ends

Lov\'{a}sz and Cherkassky discovered independently that, if $G$ is a finite graph $T\subseteq V(G)$ such that the degree $d_G(v)$ even for every vertex $v\in V(G)\setminus T$, then maximum number of edge-disjoint paths which are internally disjoint from~$T$ connect distinct vertices $T$ equal to $\frac{1}{2} \sum_{t\in T}\lambda_G(t, T\setminus \{t\})$ (where $\lambda_G(t, size smallest cut separates $t$ $T\setminus\{t\}$). From another perspective, this means $t\in in any optimal path-system there many between and~$T\setminus\{t\}$. We extend theorem based on reformulation all locally-finite infinite graphs their ends. In our generalisation, may contain not just but ends as well, one-way (two-way) when they establish vertex-end (end-end) connection.