  The challenge to obtain from the Euclidean Bethe--Salpeter amplitude the
amplitude in Minkowski is solved by resorting to un-Wick rotating the Euclidean
homogeneous integral equation. The results obtained with this new practical
method for the amputated Bethe--Salpeter amplitude for a two-boson bound state
reveals a rich analytic structure of this amplitude, which can be traced back
to the Minkowski space Bethe--Salpeter equation using the Nakanishi integral
representation. The method can be extended to small rotation angles bringing
the Euclidean solution closer to the Minkowski one and could allow in principle
the extraction of the longitudinal parton density functions and momentum
distribution amplitude, for example.
