  A well-known conjecture of Alspach says that every $2k$-regular Cayley graph
of an abelian group can be decomposed into Hamiltonian cycles. We consider an
analogous question for infinite abelian groups. In this setting one natural
analogue of a Hamiltonian cycle is a spanning double-ray. However, a naive
generalisation of Alspach's conjecture fails to hold in this setting due to the
existence of $2k$-regular Cayley graphs with finite cuts $F$ where $|F|$ and
$k$ differ in parity, which necessarily preclude the existence of a
decomposition into spanning double-rays.
  We show that every $4$-regular Cayley graph of an infinite abelian group all
of whose finite cuts are even can be decomposed into spanning double-rays, and
so characterise when such decompositions exist. We also characterise when such
graphs can be decomposed either into Hamiltonian circles, a more topological
generalisation of a Hamiltonian cycle in infinite graphs, or into a Hamiltonian
circle and a spanning double-ray.
