  It is well known that, for even n, the n-soliton solution on the Minkowski
seed, constructed using the inverse-scattering method (ISM) of Belinski and
Zakharov (BZ), is the multi-Kerr-NUT solution. We show that, for odd n, the
natural seed to use is the Euclidean space with two manifest translational
symmetries, and the n-soliton solution is the accelerating multi-Kerr-NUT
solution. We thus define the n-soliton solution on flat space for any positive
integer n. It admits both Lorentzian and Euclidean sections. In the latter
section, we find that a number, say m, of solitons can be eliminated in a
non-trivial way by appropriately fixing their corresponding so-called BZ
parameters. The resulting solutions, which may split into separate classes, are
collectively denoted as [n-m]-soliton solutions on flat space. We then carry
out a systematic study of the n- and [n-m]-soliton solutions on flat space.
This includes, in particular, an explicit presentation of their ISM
construction, an analysis of their local geometries, and a classification of
all separate classes of solutions they form. We also show how even-soliton
solutions on the seeds of the collinearly centred Gibbons-Hawking and Taub-NUT
arise from these solutions.
