  Three-magnon problem for exactly rung-dimerized spin ladder is brought up
separately at all total spin sectors. At first a special duality transformation
of the $\rm Schr\ddot odinger$ equation is found within general outlook. Then
the problem is treated within Coordinate Bethe Ansatze. A straightforward
approach is developed to obtain pure scattering states. At values S=0 and S=3
of total spin the $\rm Schr\ddot odinger$ equation has the form inherent in the
$XXZ$ chain. For $S=1,2$ solvability holds only in five previously found {\it
completely integrable} cases. Nevertheless a partial S=1 Bethe solution always
exists even for general non integrable model. Pure scattering states for all
total spin sectors are presented explicitly.
