  Formation of fermion bag solitons is an important paradigm in the theory of
hadron structure. We study this phenomenon non-perturbatively in the 1+1
dimensional Massive Gross-Neveu model, in the large $N$ limit. We find,
applying inverse scattering techniques, that the extremal static bag
configurations are reflectionless, as in the massless Gross-Neveu model. This
adds to existing results of variational calculations, which used reflectionless
bag profiles as trial configurations. Only reflectionless trial configurations
which support a single pair of charge-conjugate bound states of the associated
Dirac equation were used in those calculations, whereas the results in the
present paper hold for bag configurations which support an arbitrary number of
such pairs. We compute the masses of these multi-bound state solitons, and
prove that only bag configurations which bear a single pair of bound states are
stable. Each one of these configurations gives rise to an O(2N) antisymmetric
tensor multiplet of soliton states, as in the massless Gross-Neveu model.
