  We study axially symmetric multi-soliton solutions of a complex scalar field
theory with a sextic potential, minimally coupled to Einstein's gravity. These
solutions carry no angular momentum and can be classified by the number of
nodes of the scalar field, $k_z$, along the symmetry axis; they are interpreted
as chains with $k_z+1$ boson stars, bound by gravity, but kept apart by
repulsive scalar interactions. Chains with an odd number of constituents show a
spiraling behavior for their ADM mass (and Noether charge) in terms of their
angular frequency, similarly to a single fundamental boson star, as long as the
gravitational coupling is small; for larger coupling, however, the inner part
of the spiral is replaced by a merging with the fundamental branch of radially
excited spherical boson stars. Chains with an even number of constituents
exhibit a truncated spiral pattern, with only two or three branches, ending at
a limiting solution with finite values of ADM mass and Noether charge.
