  The natural infinite analogue of a (finite) Hamilton cycle is a
two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although
it is known that every connected $2k$-valent infinite circulant graph has a
two-way-infinite Hamilton path, there exist many such graphs that do not have a
decomposition into $k$ edge-disjoint two-way-infinite Hamilton paths. This
contrasts with the finite case where it is conjectured that every $2k$-valent
connected circulant graph has a decomposition into $k$ edge-disjoint Hamilton
cycles. We settle the problem of decomposing $2k$-valent infinite circulant
graphs into $k$ edge-disjoint two-way-infinite Hamilton paths for $k=2$, in
many cases when $k=3$, and in many other cases including where the connection
set is $\pm\{1,2,\ldots,k\}$ or $\pm\{1,2,\ldots,k-1,k+1\}$.
