  An exactly solvable effective model is suggested for an asymmetric spin
ladder with dimerized rungs. Magnon mode truncation originated from magnon
decay (recently observed in the 1D compound ${\rm IPA-CuCl}_3$) is naturally
described within this model. Using Bethe Ansatze we described a one-magnon
sector and obtained relations between interaction constants of the model and
experimentally observable quantities such as the gap and truncation energies,
spin velocity and the truncation wave vector. It is also shown that structure
factor turns to zero at the truncation point.
